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<!DOCTYPE html>
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<title>Solving Linear Systems with Two Variables</title>
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<div class="section" id="fwk-redden-ch03_s02" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">3.2</span> Solving Linear Systems with Two Variables</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch03_s02_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch03_s02_o01" numeration="arabic">
<li>Solve linear systems using the substitution method.</li>
<li>Solve linear systems using the elimination method.</li>
<li>Identify the strengths and weaknesses of each method.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch03_s02_s01" version="5.0" lang="en">
<h2 class="title editable block">The Substitution Method</h2>
<p class="para editable block" id="fwk-redden-ch03_s02_s01_p01">In this section, we review a completely algebraic technique for solving systems, the <span class="margin_term"><a class="glossterm">substitution method</a><span class="glossdef">A means of solving a linear system by solving for one of the variables and substituting the result into the other equation.</span></span>. The idea is to solve one equation for one of the variables and substitute the result into the other equation. After performing this substitution step, we are left with a single equation with one variable, which can be solved using algebra.</p>
<div class="callout block" id="fwk-redden-ch03-s02_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch03_s02_s01_p02">Solve by substitution: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0151" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>8</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p03">Solve for either variable in either equation. If you choose the first equation, you can isolate <em class="emphasis">y</em> in one step.</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p04"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0152" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s01_p05">Substitute the expression <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0153" display="inline"><mrow><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span> for the variable <em class="emphasis">y</em> in the <em class="emphasis bolditalic">other</em> equation.</p>
<div class="informalfigure large">
<img src="section_06/c4ebb8396d6fda10444882fe32dc0b8e.png">
</div>
<p class="para" id="fwk-redden-ch03_s02_s01_p07"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0154" display="block"><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mo>−</mo><mstyle color="#007f3f"><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mstyle></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>8</mn></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch03_s02_s01_p08">This leaves us with an equivalent equation with one variable, which can be solved using the techniques learned up to this point. Solve for the remaining variable.</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p09"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0155" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mstyle></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>6</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi><mo>+</mo><mn>6</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>14</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch03_s02_s01_p10"><span class="margin_term"><a class="glossterm">Back substitute</a><span class="glossdef">Once a value is found for a variable, substitute it back into one of the original equations, or its equivalent, to determine the corresponding value of the other variable.</span></span> to find the other coordinate. Substitute <em class="emphasis">x</em> = −2 into either of the original equations or their equivalents. Typically, we use the equivalent equation that we found when isolating a variable in the first step.</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p11"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0156" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s01_p12">Remember to present the solution as an ordered pair: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0157" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span>. Verify that these coordinates solve both equations of the original system:</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p13"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Check:</em> (−2, 1)</p></th>
<th></th>
</tr>
<tr>
<th align="center"><p class="para"><em class="emphasis">Equation 1</em></p></th>
<th align="center"><p class="para"><em class="emphasis">Equation 2</em></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0158" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>4</mn><mo>+</mo><mn>1</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>3</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn><mi> </mi><mi> </mi><mstyle color="#007fbf"><mi>✓</mi></mstyle></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0159" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle><mrow><mo>(</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>6</mn><mo>−</mo><mn>2</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>8</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>8</mn><mi> </mi><mi> </mi><mi> </mi><mstyle color="#007fbf"><mi>✓</mi></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch03_s02_s01_p14">The graph of this linear system follows:</p>
<div class="informalfigure large">
<img src="section_06/d96f1c69a85d456f86fcfedbddf0487c.png">
</div>
<p class="para" id="fwk-redden-ch03_s02_s01_p16">The substitution method for solving systems is a completely algebraic method. Thus graphing the lines is not required.</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p17">Answer: (−2, 1)</p>
</div>
<div class="callout block" id="fwk-redden-ch03-s02_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch03_s02_s01_p18">Solve by substitution: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0160" display="inline"><mrow><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p19">It does not matter which variable we choose to isolate first. In this case, begin by solving for <em class="emphasis">x</em> in the first equation.</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p20"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0161" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn><mi>y</mi><mo>+</mo><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>5</mn><mi>y</mi><mo>+</mo><mn>9</mn></mrow><mn>3</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>y</mi><mo>+</mo><mn>3</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s01_p21"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0162" display="block"><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr columnalign="right"><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi></mrow><mo>=</mo><mn>9</mn></mtd><mtd columnalign="right"><mtext> </mtext><mtext> </mtext><mo>⇒</mo><mtext> </mtext><mtext> </mtext></mtd><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mi>x</mi></mstyle><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>y</mi><mo>+</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>4</mn><mstyle color="#007fbf"><mi>x</mi></mstyle><mo>+</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow></mtd><mtd></mtd><mtd></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s01_p22">Next, substitute into the second equation and solve for <em class="emphasis">y</em>.</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p23"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0163" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>4</mn><mrow><mo>(</mo><mrow><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>20</mn></mrow><mn>3</mn></mfrac><mi>y</mi><mo>+</mo><mn>12</mn><mo>+</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mrow><mn>26</mn></mrow><mn>3</mn></mfrac><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>13</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>13</mn><mrow><mo>(</mo><mrow><mfrac><mn>3</mn><mrow><mn>26</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s01_p24">Back substitute into the equation used in the substitution step:</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p25"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0164" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>y</mi><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>5</mn><mn>3</mn></mfrac><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s01_p26">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0165" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch03-s02_s01_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch03_s02_s01_p27"><strong class="emphasis bold">Try this!</strong> Solve by substitution: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0166" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p28">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0167" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mtext> </mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p>
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</div>
</div>
<p class="para editable block" id="fwk-redden-ch03_s02_s01_p30">As we know, not all linear systems have only one ordered pair solution. Next, we explore what happens when using the substitution method to solve a dependent system.</p>
<div class="callout block" id="fwk-redden-ch03-s02_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch03_s02_s01_p31">Solve by substitution: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0168" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>10</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p32">Since the first equation has a term with coefficient 1, we choose to solve for that first.</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p33"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0169" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mover><mo>⇒</mo><mrow> </mrow></mover><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>y</mi></mstyle><mo>=</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>10</mn><mi>x</mi><mo>−</mo><mn>2</mn><mstyle color="#007fbf"><mi>y</mi></mstyle></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr></mtable></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s01_p34">Next, substitute this expression in for <em class="emphasis">y</em> in the second equation.</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p35"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0170" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>10</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>10</mn><mi>x</mi><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>10</mn><mi>x</mi><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>2</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mi> </mi><mi> </mi><mi> </mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi> </mi><mstyle color="#007fbf"><mi>T</mi><mi>r</mi><mi>u</mi><mi>e</mi></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s01_p36">This process led to a true statement; hence the equation is an identity and any real number is a solution. This indicates that the system is dependent. The simultaneous solutions take the form <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0171" display="inline"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></math></span>, or in this case, <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0172" display="inline"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi> </mi><mtext> </mtext><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math></span>, where <em class="emphasis">x</em> is any real number.</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p37">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0173" display="inline"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi> </mi><mtext> </mtext><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch03_s02_s01_p38">To have a better understanding of the previous example, rewrite both equations in slope-intercept form and graph them on the same set of axes.</p>
<p class="para block" id="fwk-redden-ch03_s02_s01_p39"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0174" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>10</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr></mtable></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mover><mo>⇒</mo><mrow> </mrow></mover><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow></math></span></p>
<div class="informalfigure large block">
<img src="section_06/4995c4539f20276813b2ee0d2514d06f.png">
</div>
<p class="para editable block" id="fwk-redden-ch03_s02_s01_p41">We can see that both equations represent the same line, and thus the system is dependent. Now explore what happens when solving an inconsistent system using the substitution method.</p>
<div class="callout block" id="fwk-redden-ch03-s02_s01_n04">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch03_s02_s01_p42">Solve by substitution: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0175" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>14</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>16</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p43">Solve for <em class="emphasis">y</em> in the first equation.</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p44"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0176" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mn>3</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>7</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s01_p45"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0177" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mover><mo>⇒</mo><mrow> </mrow></mover><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>y</mi></mstyle><mo>=</mo><mfrac><mn>7</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>14</mn><mi>x</mi><mo>−</mo><mn>6</mn><mstyle color="#007fbf"><mi>y</mi></mstyle></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>16</mn></mtd></mtr></mtable></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s01_p46">Substitute into the second equation and solve.</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p47"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0178" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>14</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>16</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>14</mn><mi>x</mi><mo>−</mo><mn>6</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mfrac><mn>7</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>16</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>14</mn><mi>x</mi><mo>−</mo><mover><mrow><menclose notation="updiagonalstrike"><mn>6</mn></menclose></mrow><mstyle color="#007fbf"><mn>2</mn></mstyle></mover><mo>⋅</mo><mfrac><mn>7</mn><mrow><munder><mrow><menclose notation="updiagonalstrike"><mn>3</mn></menclose></mrow><mstyle color="#007fbf"><mn>1</mn></mstyle></munder></mrow></mfrac><mi>x</mi><mo>−</mo><mn>6</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>16</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>14</mn><mi>x</mi><mo>−</mo><mn>14</mn><mi>x</mi><mo>−</mo><mn>6</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>16</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>6</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>16</mn><mi> </mi><mi> </mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi> </mi><mstyle color="#ff0000"><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s01_p48">Solving leads to a false statement. This indicates that the equation is a contradiction. There is no solution for <em class="emphasis">x</em> and hence no solution to the system.</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p49">Answer: Ø</p>
</div>
<p class="para editable block" id="fwk-redden-ch03_s02_s01_p50">A false statement indicates that the system is inconsistent, or in geometric terms, that the lines are parallel and do not intersect. To illustrate this, determine the slope-intercept form of each line and graph them on the same set of axes.</p>
<p class="para block" id="fwk-redden-ch03_s02_s01_p51"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0179" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>14</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>16</mn></mtd></mtr></mtable></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mover><mo>⇒</mo><mrow> </mrow></mover><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>7</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>7</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>8</mn><mn>3</mn></mfrac></mtd></mtr></mtable></mrow></mrow></math></span></p>
<div class="informalfigure large block">
<img src="section_06/29f0a74d32c346db54accdfa033095c0.png">
</div>
<p class="para editable block" id="fwk-redden-ch03_s02_s01_p53">In slope-intercept form, it is easy to see that the two lines have the same slope but different <em class="emphasis">y</em>-intercepts.</p>
<div class="callout block" id="fwk-redden-ch03-s02_s01_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch03_s02_s01_p54"><strong class="emphasis bold">Try this!</strong> Solve by substitution: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0180" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>4</mn><mi>x</mi><mo>−</mo><mn>10</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="para" id="fwk-redden-ch03_s02_s01_p55">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0181" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mtext> </mtext><mtext> </mtext><mfrac><mn>2</mn><mn>5</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>3</mn><mn>5</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p>
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<div class="section" id="fwk-redden-ch03_s02_s02" version="5.0" lang="en">
<h2 class="title editable block">The Elimination Method</h2>
<p class="para editable block" id="fwk-redden-ch03_s02_s02_p01">In this section, the goal is to review another completely algebraic method for solving a system of linear equations called the <span class="margin_term"><a class="glossterm">elimination method</a><span class="glossdef">A means of solving a system by adding equivalent equations in such a way as to eliminate a variable.</span></span> or <span class="margin_term"><a class="glossterm">addition method</a><span class="glossdef">Often used when referring to the elimination method for solving systems.</span></span>. This method depends on the <span class="margin_term"><a class="glossterm">addition property of equations</a><span class="glossdef">If <em class="emphasis">A</em>, <em class="emphasis">B</em>, <em class="emphasis">C</em>, and <em class="emphasis">D</em> are algebraic expressions, where <em class="emphasis">A</em> = <em class="emphasis">B</em> and <em class="emphasis">C</em> = <em class="emphasis">D</em>, then <em class="emphasis">A</em> + <em class="emphasis">C</em> = <em class="emphasis">B</em> + <em class="emphasis">D</em>.</span></span>: given algebraic expressions <em class="emphasis">A</em>, <em class="emphasis">B</em>, <em class="emphasis">C</em>, and <em class="emphasis">D</em> we have</p>
<p class="para block" id="fwk-redden-ch03_s02_s02_p02"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0182" display="block"><mrow><mtext>If</mtext><mi> </mi><mi> </mi><mi>A</mi><mo>=</mo><mi>B</mi><mi> </mi><mi> </mi><mtext>and</mtext><mi> </mi><mi>C</mi><mo>=</mo><mi>D</mi><mo>,</mo><mi> </mi><mi> </mi><mtext>then</mtext><mi> </mi><mtext> </mtext><mi>A</mi><mo>+</mo><mi>C</mi><mo>=</mo><mi>B</mi><mo>+</mo><mi>D</mi></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch03_s02_s02_p03">Consider the following system:</p>
<p class="para block" id="fwk-redden-ch03_s02_s02_p04"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0183" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mi> </mi><mi>x</mi><mo>−</mo><mi>y</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch03_s02_s02_p05">We can add the equations together to eliminate the variable <em class="emphasis">y</em>.</p>
<p class="para block" id="fwk-redden-ch03_s02_s02_p06"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0184" display="block"><mtable columnspacing="0.1em" rowlines="none solid none"><mtr><mtd></mtd><mtd><mi>x</mi><mstyle color="#ff0000"><mtext> </mtext><mo>+</mo><mtext> </mtext><mi>y</mi></mstyle></mtd><mtd><mo>=</mo></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd style="border-bottom:1pt solid black"><mo>+</mo></mtd><mtd style="border-bottom:1pt solid black"><mi>x</mi><mstyle color="#ff0000"><mo>−</mo><mi>y</mi></mstyle></mtd><mtd style="border-bottom:1pt solid black"><mo>=</mo></mtd><mtd style="border-bottom:1pt solid black"><mn>1</mn></mtd></mtr><mtr><mtd></mtd><mtd columnalign="left"><mn>2</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>6</mn></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch03_s02_s02_p07">This leaves us with a linear equation with one variable that can be easily solved:</p>
<p class="para block" id="fwk-redden-ch03_s02_s02_p08"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0185" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch03_s02_s02_p09">At this point, we have the <em class="emphasis">x</em>-coordinate of the simultaneous solution, so all that is left to do is back substitute to find the corresponding <em class="emphasis">y</em>-value.</p>
<p class="para block" id="fwk-redden-ch03_s02_s02_p10"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0186" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mstyle color="#007f3f"><mn>3</mn></mstyle><mo>+</mo><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch03_s02_s02_p11">The solution to the system is (3, 2). Of course, the variable is not always so easily eliminated. Typically, we have to find an equivalent system by applying the multiplication property of equality to one or both of the equations as a means to line up one of the variables to eliminate. The goal is to arrange that either the <em class="emphasis">x</em> terms or the <em class="emphasis">y</em> terms are opposites, so that when the equations are added, the terms eliminate.</p>
<div class="callout block" id="fwk-redden-ch03-s02_s02_n01">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch03_s02_s02_p12">Solve by elimination: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0187" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></mtd><mtd columnalign="right"><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p13">We choose to eliminate the terms with variable <em class="emphasis">y</em> because the coefficients have different signs. To do this, we first determine the least common multiple of the coefficients; in this case, the LCM(3, 2) is 6. Therefore, multiply both sides of both equations by the appropriate values to obtain coefficients of −6 and 6. This results in the following equivalent system:</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p14"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0188" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn><mtext> </mtext><mtext> </mtext></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd></mtr></mtable></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mover><mo>⇒</mo><mrow><mo>×</mo><mn>2</mn></mrow></mover></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><munder><mo>⇒</mo><mrow><mo>×</mo><mn>3</mn></mrow></munder></mrow></mtd></mtr></mtable><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>10</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>21</mn></mtd></mtr></mtable></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s02_p15">The terms involving <em class="emphasis">y</em> are now lined up to eliminate. Add the equations together and solve for <em class="emphasis">x</em>.</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p16"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0189" display="block"><mtable columnspacing="0.1em" rowlines="none solid none none"><mtr><mtd columnalign="right"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>10</mn><mi>x</mi><mstyle color="#ff0000"><mo>−</mo><mn>6</mn><mi>y</mi></mstyle></mtd><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mtext> </mtext><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>9</mn><mi>x</mi><mstyle color="#ff0000"><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>6</mn><mi>y</mi></mstyle></mtd><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>21</mn></mtd></mtr><mtr><mtd columnalign="right"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>19</mn><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>19</mn></mtd></mtr><mtr><mtd columnalign="right"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>x</mi></mtd><mtd></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s02_p17">Back substitute.</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p18"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0190" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mo>+</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s02_p19">Therefore the simultaneous solution is (1, 2). The check follows.</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p20"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Check:</em> (1, 2)</p></th>
<th></th>
</tr>
<tr>
<th align="center"><p class="para"><em class="emphasis">Equation 1:</em></p></th>
<th align="center"><p class="para"><em class="emphasis">Equation 2:</em></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0191" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>3</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn><mo>−</mo><mn>6</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>1</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn><mi> </mi><mi> </mi><mi> </mi><mstyle color="#007fbf"><mi>✓</mi></mstyle></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0192" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mo>+</mo><mn>4</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>7</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn><mi> </mi><mi> </mi><mi> </mi><mstyle color="#007fbf"><mi>✓</mi></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch03_s02_s02_p21">Answer: (1, 2)</p>
</div>
<p class="para block" id="fwk-redden-ch03_s02_s02_p22">Sometimes linear systems are not given in standard form <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0193" display="inline"><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>=</mo><mi>c</mi></mrow></math></span>. When this is the case, it is best to rearrange the equations before beginning the steps to solve by elimination. Also, we can eliminate either variable. The goal is to obtain a solution for one of the variables and then back substitute to find a solution for the other.</p>
<div class="callout block" id="fwk-redden-ch03-s02_s02_n02">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch03_s02_s02_p23">Solve by elimination: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0194" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="right"><mn>12</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>11</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><mi>y</mi><mo>+</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p24">First, rewrite the second equation in standard form.</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p25"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0195" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><mi>y</mi><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s02_p26">This results in an equivalent system in standard form, where like terms are aligned in columns.</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p27"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0196" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="right"><mn>12</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>11</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><mi>y</mi><mo>+</mo><mn>1</mn></mtd></mtr></mtable></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mover><mo>⇒</mo><mrow> </mrow></mover><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>12</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>11</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s02_p28">We can eliminate the term with variable <em class="emphasis">x</em> if we multiply the second equation by −4.</p>
<div class="informalfigure large">
<img src="section_06/c1c845246da68b14739fc1ec5922f938.png">
</div>
<p class="para" id="fwk-redden-ch03_s02_s02_p30">Next, we add the equations together,</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p31"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0197" display="block"><mtable columnspacing="0.1em" rowlines="none solid none none"><mtr><mtd columnalign="right"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#ff0000"><mn>12</mn><mi>x</mi></mstyle><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>11</mn></mtd></mtr><mtr><mtd columnalign="right"><mtext> </mtext><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#ff0000"><mo>−</mo><mn>12</mn><mi>x</mi></mstyle><mo>+</mo><mn>16</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>4</mn></mtd></mtr><mtr><mtd columnalign="right"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>21</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>7</mn><mrow><mn>21</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s02_p32">Back substitute.</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p33"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0198" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><mi>y</mi><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><mrow><mo>(</mo><mrow><mfrac><mstyle color="#007f3f"><mn>1</mn></mstyle><mstyle color="#007f3f"><mn>3</mn></mstyle></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>7</mn><mn>3</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>7</mn><mn>3</mn></mfrac><mo>⋅</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>7</mn><mn>9</mn></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s02_p34">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0199" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>7</mn><mn>9</mn></mfrac><mo>,</mo><mtext> </mtext><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch03-s02_s02_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch03_s02_s02_p35"><strong class="emphasis bold">Try this!</strong> Solve by elimination: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0200" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>9</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p36">Answer: (−5, 3)</p>
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<a data-iframe-code='<iframe src="http://www.youtube.com/v/FX90hfggjbI" condition="http://img.youtube.com/vi/FX90hfggjbI/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/FX90hfggjbI" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<p class="para editable block" id="fwk-redden-ch03_s02_s02_p38">At this point, we explore what happens when solving dependent and inconsistent systems using the elimination method.</p>
<div class="callout block" id="fwk-redden-ch03-s02_s02_n03">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch03_s02_s02_p39">Solve by elimination: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0201" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>6</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>14</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p40">To eliminate the variable <em class="emphasis">x</em>, we could multiply the first equation by −2.</p>
<div class="informalfigure large">
<img src="section_06/0e355e8f73169140ce519a886e768fbe.png">
</div>
<p class="para" id="fwk-redden-ch03_s02_s02_p42">Now adding the equations we have</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p43"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0202" display="block"><mrow><mtable columnspacing="0.1em" rowlines="none solid none"><mtr><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mstyle color="#ff0000"><mn>2</mn><mi>y</mi></mstyle></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>14</mn></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="left" style="border-bottom:1pt solid black"><mo>+</mo></mtd><mtd columnalign="right" style="border-bottom:1pt solid black"><mrow><mn>6</mn><mi>x</mi><mo>−</mo><mstyle color="#ff0000"><mn>2</mn><mi>y</mi></mstyle></mrow></mtd><mtd style="border-bottom:1pt solid black"><mo>=</mo></mtd><mtd columnalign="left" style="border-bottom:1pt solid black"><mrow><mn>14</mn></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd columnalign="right"><mn>0</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="right"><mrow><mtext> </mtext><mstyle color="#007fbf"><mi>T</mi><mi>r</mi><mi>u</mi><mi>e</mi></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s02_p44">A true statement indicates that this is a dependent system. The lines coincide, and we need <em class="emphasis">y</em> in terms of <em class="emphasis">x</em> to present the solution set in the form <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0203" display="inline"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mtext> </mtext><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></math></span>. Choose one of the original equations and solve for <em class="emphasis">y</em>. Since the equations are equivalent, it does not matter which one we choose.</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p45"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0204" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>−</mo><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></mtd></mtr><mtr><mtd columnalign="right"><mstyle color="#007fbf"><mo>−</mo><mn>1</mn></mstyle><mrow><mo>(</mo><mrow><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>−</mo><mn>1</mn></mstyle><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>−</mo><mn>7</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s02_p46">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0205" display="inline"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi> </mi><mtext> </mtext><mn>3</mn><mi>x</mi><mo>−</mo><mn>7</mn><mo stretchy="false">)</mo></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch03-s02_s02_n03a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch03_s02_s02_p47"><strong class="emphasis bold">Try this!</strong> Solve by elimination: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0206" display="inline"><mrow><mi> </mi><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>15</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>15</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>30</mn></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p48">Answer: No solution, <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0207" display="inline"><mo>Ø</mo></math></span></p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/E4B0lMLEliY" condition="http://img.youtube.com/vi/E4B0lMLEliY/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/E4B0lMLEliY" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<p class="para editable block" id="fwk-redden-ch03_s02_s02_p50">Given a linear system where the equations have fractional coefficients, it is usually best to clear the fractions before beginning the elimination method.</p>
<div class="callout block" id="fwk-redden-ch03-s02_s02_n04">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch03_s02_s02_p51">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0208" display="inline"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mfrac><mn>1</mn><mrow><mn>10</mn></mrow></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>4</mn><mn>5</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi> </mi><mfrac><mn>1</mn><mn>7</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>2</mn><mrow><mn>21</mn></mrow></mfrac></mtd></mtr></mtable></mrow></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p52">Recall that we can clear fractions by multiplying both sides of an equation by the least common multiple of the denominators (LCD). Take care to distribute and then simplify.</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p53"></p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para"><em class="emphasis">Equation 1</em></p></td>
<td align="center"><p class="para"><em class="emphasis">Equation 2</em></p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0209" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mstyle color="#007fbf"><mn>10</mn></mstyle><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>1</mn><mrow><mn>10</mn></mrow></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>10</mn></mstyle><mrow><mo>(</mo><mrow><mfrac><mn>4</mn><mn>5</mn></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mstyle color="#007fbf"><mn>10</mn></mstyle><mo>⋅</mo><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>1</mn><mrow><mn>10</mn></mrow></mfrac><mi>x</mi></mrow><mo>)</mo></mrow><mo>+</mo><mstyle color="#007fbf"><mn>10</mn></mstyle><mo>⋅</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>10</mn></mstyle><mo>⋅</mo><mfrac><mn>4</mn><mn>5</mn></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0210" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mstyle color="#007fbf"><mn>21</mn></mstyle><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>7</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>y</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>21</mn></mstyle><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>2</mn><mrow><mn>21</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mstyle color="#007fbf"><mn>21</mn></mstyle><mo>⋅</mo><mfrac><mn>1</mn><mn>7</mn></mfrac><mi>x</mi><mo>+</mo><mstyle color="#007fbf"><mn>21</mn></mstyle><mo>⋅</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>21</mn></mstyle><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>2</mn><mrow><mn>21</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch03_s02_s02_p54">This results in an equivalent system where the equations have integer coefficients,</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p55"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0211" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mfrac><mn>1</mn><mrow><mn>10</mn></mrow></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>4</mn><mn>5</mn></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd></mtr><mtr><mtd columnalign="right"><mi> </mi><mfrac><mn>1</mn><mn>7</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>2</mn><mrow><mn>21</mn></mrow></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd></mtr></mtable></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mover><mo>⇒</mo><mrow><mo>×</mo><mn>10</mn></mrow></mover></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><munder><mo>⇒</mo><mrow><mo>×</mo><mn>21</mn></mrow></munder></mrow></mtd></mtr></mtable><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi> </mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd></mtr><mtr><mtd columnalign="right"><mi> </mi><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr></mtable></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s02_p56">Solve using the elimination method.</p>
<div class="informalfigure large">
<img src="section_06/f199eea33ed4bb7832221e93c546f85f.png">
</div>
<p class="para" id="fwk-redden-ch03_s02_s02_p58"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0212" display="block"><mrow><mtable columnspacing="0.1em" rowlines="none solid none none"><mtr><mtd><mrow></mrow></mtd><mtd columnalign="right"><mrow><mstyle color="#ff0000"><mo>−</mo><mn>3</mn><mi>x</mi></mstyle><mtext> </mtext><mo>+</mo><mn>15</mn><mi>y</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>24</mn></mrow></mtd></mtr><mtr><mtd columnalign="left" style="border-bottom:1px solid black"><mo>+</mo></mtd><mtd columnalign="right" style="border-bottom:1px solid black"><mrow><mstyle color="#ff0000"><mn>3</mn><mi>x</mi></mstyle><mtext> </mtext><mo>+</mo><mn>7</mn><mi>y</mi></mrow></mtd><mtd style="border-bottom:1px solid black"><mo>=</mo></mtd><mtd columnalign="left" style="border-bottom:1px solid black"><mrow><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>22</mn><mi>y</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>22</mn></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s02_p59">Back substitute.</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p60"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0213" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>3</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch03_s02_s02_p61">Answer: (−3, 1)</p>
</div>
<p class="para editable block" id="fwk-redden-ch03_s02_s02_p62">We can use a similar technique to clear decimals before solving.</p>
<div class="callout block" id="fwk-redden-ch03-s02_s02_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch03_s02_s02_p63"><strong class="emphasis bold">Try this!</strong> Solve using elimination: <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0214" display="inline"><mrow><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>y</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>8</mn><mn>3</mn></mfrac></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>.</p>
<p class="para" id="fwk-redden-ch03_s02_s02_p64">Answer: (5, −2)</p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/ujlpeP7nakE" condition="http://img.youtube.com/vi/ujlpeP7nakE/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/ujlpeP7nakE" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
</div>
<div class="section" id="fwk-redden-ch03_s02_s03" version="5.0" lang="en">
<h2 class="title editable block">Summary of the Methods for Solving Linear Systems</h2>
<p class="para editable block" id="fwk-redden-ch03_s02_s03_p01">We have reviewed three methods for solving linear systems of two equations with two variables. Each method is valid and can produce the same correct result. In this section, we summarize the strengths and weaknesses of each method.</p>
<p class="para editable block" id="fwk-redden-ch03_s02_s03_p02">The graphing method is useful for understanding what a system of equations is and what the solutions must look like. When the equations of a system are graphed on the same set of axes, we can see that the solution is the point where the graphs intersect. The graphing is made easy when the equations are in slope-intercept form. For example,</p>
<p class="para block" id="fwk-redden-ch03_s02_s03_p03"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0215" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn><mi>x</mi><mo>+</mo><mn>15</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>5</mn></mtd></mtr></mtable></mrow></mrow></math></span></p>
<div class="informalfigure large block">
<img src="section_06/343553149db59973f7cf3eba035c0c1a.png">
</div>
<p class="para editable block" id="fwk-redden-ch03_s02_s03_p05">The simultaneous solution (−1, 10) corresponds to the point of intersection. One drawback of this method is that it is very inaccurate. When the coordinates of the solution are not integers, the method is practically unusable. If we have a choice, we typically avoid this method in favor of the more accurate algebraic techniques.</p>
<p class="para editable block" id="fwk-redden-ch03_s02_s03_p06">The substitution method, on the other hand, is a completely algebraic method. It requires you to solve for one of the variables and substitute the result into the other equation. The resulting equation has one variable for which you can solve. This method is particularly useful when there is a variable within the system with coefficient of 1. For example,</p>
<p class="para block" id="fwk-redden-ch03_s02_s03_p07"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0216" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>10</mn><mi>x</mi><mo>+</mo><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>20</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>7</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>14</mn></mtd></mtr></mtable></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>o</mi><mi>o</mi><mi>s</mi><mi>e</mi><mi> </mi><mi>t</mi><mi>h</mi><mi>e</mi><mi> </mi><mi>s</mi><mi>u</mi><mi>b</mi><mi>s</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi> </mi><mi>m</mi><mi>e</mi><mi>t</mi><mi>h</mi><mi>o</mi><mi>d</mi><mtext>.</mtext></mstyle></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch03_s02_s03_p08">In this case, it is easy to solve for <em class="emphasis">y</em> in the first equation and then substitute the result into the other equation. One drawback of this method is that it often leads to equivalent equations with fractional coefficients, which are tedious to work with. If there is not a coefficient of 1, then it usually is best to choose the elimination method.</p>
<p class="para editable block" id="fwk-redden-ch03_s02_s03_p09">The elimination method is a completely algebraic method which makes use of the addition property of equations. We multiply one or both of the equations to obtain equivalent equations where one of the variables is eliminated if we add them together. For example,</p>
<p class="para block" id="fwk-redden-ch03_s02_s03_p10"><span class="informalequation"><math xml:id="fwk-redden-ch03_m0217" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>5</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>16</mn></mtd></mtr></mtable></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>o</mi><mi>o</mi><mi>s</mi><mi>e</mi><mi> </mi><mi>t</mi><mi>h</mi><mi>e</mi><mi> </mi><mi>e</mi><mi>l</mi><mi>i</mi><mi>m</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi> </mi><mi>m</mi><mi>e</mi><mi>t</mi><mi>h</mi><mi>o</mi><mi>d</mi><mtext>.</mtext></mstyle></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch03_s02_s03_p11">To eliminate the terms involving <em class="emphasis">x</em>, we would multiply both sides of the first equation by 5 and both sides of the second equation by −2. This results in an equivalent system where the variable <em class="emphasis">x</em> is eliminated when we add the equations together. Of course, there are other combinations of numbers that achieve the same result. We could even choose to eliminate the variable <em class="emphasis">y</em>. No matter which variable is eliminated first, the solution will be the same. Note that the substitution method, in this case, would require tedious calculations with fractional coefficients. One weakness of the elimination method, as we will see later in our study of algebra, is that it does not always work for nonlinear systems.</p>
<div class="key_takeaways block" id="fwk-redden-ch03_s02_s03_n01">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch03_s02_s03_l01" mark="bullet">
<li>The substitution method requires that we solve for one of the variables and then substitute the result into the other equation. After performing the substitution step, the resulting equation has one variable and can be solved using the techniques learned up to this point.</li>
<li>The elimination method is another completely algebraic method for solving a system of equations. Multiply one or both of the equations in a system by certain numbers to obtain an equivalent system where at least one variable in both equations have opposite coefficients. Adding these equivalent equations together eliminates that variable, and the resulting equation has one variable for which you can solve.</li>
<li>It is a good practice to first rewrite the equations in standard form before beginning the elimination method.</li>
<li>Solutions to systems of two linear equations with two variables, if they exist, are ordered pairs (<em class="emphasis">x</em>, <em class="emphasis">y</em>).</li>
<li>If the process of solving a system of equations leads to a false statement, then the system is inconsistent and there is no solution, <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0218" display="inline"><mo>Ø</mo></math></span>.</li>
<li>If the process of solving a system of equations leads to an identity, then the system is dependent and there are infinitely many solutions that can be expressed using the form <span class="inlineequation"><math xml:id="fwk-redden-ch03_m0219" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow></math></span>.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch03_s02_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch03_s02_qs01_qd01">
<h3 class="title">Part A: Substitution Method</h3>
<ol class="qandadiv" id="fwk-redden-ch03_s02_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch03_s02_qs01_p01"><strong class="emphasis bold">Solve by substitution.</strong></p>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa01">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0220" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="left"><mi>y</mi><mo>=</mo><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>4</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>41</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa02">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0221" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>x</mi><mo>=</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>8</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa03">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0222" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>y</mi><mo>=</mo><mi>x</mi></mtd></mtr><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>10</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa04">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0223" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="left"><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mtd></mtr><mtr><mtd columnalign="left"><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi><mo>=</mo><mn>4</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa05">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0225" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>2</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa06">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0227" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>5</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa07">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0229" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="left"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mo>−</mo><mn>3</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa08">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0231" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="left"><mi>y</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>6</mn><mi>x</mi><mo>−</mo><mn>9</mn><mi>y</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa09">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0233" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd columnalign="left"><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mo>−</mo><mn>6</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa10">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0234" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="left"><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>7</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>=</mo><mn>9</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa11">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0236" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>=</mo><mn>19</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa12">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0237" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="left"><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa13">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0238" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>2</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>17</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa14">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0239" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>11</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>5</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa15">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0240" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa16">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0242" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>5</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mn>12</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>9</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mn>10</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa17">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0244" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>6</mn></mtd></mtr><mtr><mtd columnalign="left"><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>=</mo><mn>24</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa18">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0246" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>6</mn></mtd></mtr><mtr><mtd columnalign="left"><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>12</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa19">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0248" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="left"><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mo>−</mo><mn>4</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>6</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa20">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0250" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>10</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>−</mo><mn>10</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>20</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa21">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0252" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mn>9</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>4</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa22">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0253" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>4</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa23">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0254" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>4</mn><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa24">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0256" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>−</mo><mn>7</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>6</mn><mi>x</mi><mo>+</mo><mn>14</mn><mi>y</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa25">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0258" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>10</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="left"><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa26">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0260" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mi>y</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mtd></mtr><mtr><mtd columnalign="left"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa27">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0261" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>y</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="left"><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mn>12</mn></mrow></mfrac></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa28">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0262" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mfrac><mn>1</mn><mn>7</mn></mfrac><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd columnalign="left"><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa29">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0264" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mfrac><mn>3</mn><mn>5</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>2</mn><mn>5</mn></mfrac><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd columnalign="left"><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mrow><mn>12</mn></mrow></mfrac><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa30">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0266" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>y</mi></mtd></mtr><mtr><mtd columnalign="left"><mi>x</mi><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>y</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa31">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0267" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi><mo>=</mo><mfrac><mn>5</mn><mn>8</mn></mfrac></mtd></mtr><mtr><mtd columnalign="left"><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa32">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0269" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd columnalign="left"><mo>−</mo><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa33">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0270" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>y</mi><mo>=</mo><mn>3</mn><mi>x</mi></mtd></mtr><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa34">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0271" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>20</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>+</mo><mn>8</mn><mi>y</mi><mo>=</mo><mn>8</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa35">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0272" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>7</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa36">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0273" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn><mi>y</mi><mo>=</mo><mn>2</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>+</mo><mn>7</mn><mi>y</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa37">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0275" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>x</mi><mo>=</mo><mn>5</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa38">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0277" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mn>4</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>5</mn><mi>y</mi><mo>=</mo><mn>20</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch03_s02_qs01_qd02">
<h3 class="title">Part B: Elimination Method</h3>
<ol class="qandadiv" id="fwk-redden-ch03_s02_qs01_qd02_qd01" start="39">
<p class="para" id="fwk-redden-ch03_s02_qs01_p78"><strong class="emphasis bold">Solve by elimination.</strong></p>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa39">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0279" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="left"><mn>6</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa40">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0281" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mn>9</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa41">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0282" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mo>−</mo><mn>6</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>5</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mo>−</mo><mn>18</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa42">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0283" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="left"><mo>−</mo><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa43">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0284" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd columnalign="left"><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mo>−</mo><mn>7</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa44">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0285" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>2</mn></mtd></mtr><mtr><mtd columnalign="left"><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mo>−</mo><mn>3</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa45">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0287" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>6</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa46">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0289" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>5</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>10</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>=</mo><mn>4</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa47">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0291" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>14</mn><mi>y</mi><mo>=</mo><mn>28</mn></mtd></mtr><mtr><mtd columnalign="left"><mi>x</mi><mo>−</mo><mn>7</mn><mi>y</mi><mo>=</mo><mn>21</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa48">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0293" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>12</mn><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>24</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa49">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0295" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>x</mi><mo>+</mo><mn>8</mn><mi>y</mi><mo>=</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>+</mo><mn>12</mn><mi>y</mi><mo>=</mo><mn>6</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa50">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0297" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>15</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>4</mn><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mo>=</mo><mn>14</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa51">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0298" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>4</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>10</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>−</mo><mn>9</mn><mi>y</mi><mo>=</mo><mn>15</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa52">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0299" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>8</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>15</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa53">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0300" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>7</mn><mi>y</mi><mo>=</mo><mn>56</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>4</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>112</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa54">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0301" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mn>9</mn><mi>x</mi><mo>−</mo><mn>15</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>15</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>10</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa55">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0303" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>6</mn><mi>x</mi><mo>−</mo><mn>7</mn><mi>y</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>7</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa56">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0305" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>4</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd columnalign="left"><mo>−</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>7</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa57">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0306" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>7</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa58">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0307" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>9</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>14</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa59">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0308" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>9</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>7</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>15</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa60">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0309" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>7</mn></mtd></mtr><mtr><mtd columnalign="left"><mo>−</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi><mo>=</mo><mn>11</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa61">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0311" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>+</mo><mn>9</mn><mi>y</mi><mo>=</mo><mn>8</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa62">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0312" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>5</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa63">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0314" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>12</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>8</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa64">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0316" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>25</mn><mi>x</mi><mo>+</mo><mn>15</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>15</mn><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa65">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0318" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>2</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>18</mn><mi>x</mi><mo>−</mo><mn>12</mn><mi>y</mi><mo>=</mo><mn>5</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa66">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0320" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>4</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa67">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0321" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="left"><mn>28</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>y</mi><mo>=</mo><mn>9</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>6</mn><mi>y</mi><mo>=</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>15</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa68">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0323" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>y</mi><mo>=</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>15</mn></mtd></mtr><mtr><mtd columnalign="left"><mi>y</mi><mo>=</mo><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>5</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa69">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0324" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>9</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>5</mn><mi>x</mi><mo>−</mo><mn>8</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>16</mn></mtd></mtr></mtable></mrow><mtext> </mtext></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa70">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0325" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></mtd></mtr><mtr><mtd columnalign="left"><mfrac><mn>5</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa71">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0326" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>9</mn></mfrac><mi>y</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="left"><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa72">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0328" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mtd></mtr><mtr><mtd columnalign="left"><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mrow><mn>19</mn></mrow><mn>6</mn></mfrac></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa73">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0330" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mfrac><mrow><mn>14</mn></mrow><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd columnalign="left"><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>7</mn></mfrac><mi>y</mi><mo>=</mo><mfrac><mn>4</mn><mrow><mn>21</mn></mrow></mfrac></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa74">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0332" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>0.025</mn><mi>x</mi><mo>+</mo><mn>0.1</mn><mi>y</mi><mo>=</mo><mn>0.5</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>0.11</mn><mi>x</mi><mo>+</mo><mn>0.04</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>0.2</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa75">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0333" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>1.3</mn><mi>x</mi><mo>+</mo><mn>0.1</mn><mi>y</mi><mo>=</mo><mn>0.35</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>0.5</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mo>−</mo><mn>2.75</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa76">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0334" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>0.02</mn><mi>x</mi><mo>+</mo><mn>0.03</mn><mi>y</mi><mo>=</mo><mn>0.125</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch03_s02_qs01_qd03">
<h3 class="title">Part C: Mixed Practice</h3>
<ol class="qandadiv" id="fwk-redden-ch03_s02_qs01_qd03_qd01" start="77">
<p class="para" id="fwk-redden-ch03_s02_qs01_p155"><strong class="emphasis bold">Solve using any method.</strong></p>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa77">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0335" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="left"><mn>6</mn><mi>x</mi><mo>=</mo><mn>12</mn><mi>y</mi><mo>+</mo><mn>7</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>6</mn><mi>x</mi><mo>+</mo><mn>24</mn><mi>y</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa78">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0337" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="left"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>12</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa79">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0338" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>5</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa80">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0340" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mi>x</mi><mo>=</mo><mo>−</mo><mn>4</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa81">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0341" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mi>x</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa82">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0343" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mi>x</mi></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mo>−</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>1</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa83">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0344" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mn>5</mn><mi>x</mi></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>10</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa84">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0345" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mo>−</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>3</mn><mi>x</mi></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa85">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0347" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mn>7</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>7</mn><mi>x</mi><mo>=</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>23</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa86">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0348" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>9</mn><mi>y</mi><mo>−</mo><mn>14</mn><mo>=</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa87">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0349" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>5</mn><mrow><mn>16</mn></mrow></mfrac><mi>x</mi><mo>+</mo><mn>10</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mfrac><mn>5</mn><mrow><mn>16</mn></mrow></mfrac><mi>x</mi><mo>−</mo><mn>10</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa88">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0350" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>6</mn><mn>5</mn></mfrac><mi>x</mi><mo>+</mo><mn>12</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mi>x</mi><mo>=</mo><mn>6</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa89">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0352" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>+</mo><mi>y</mi><mo>=</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>15</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa90">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0354" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mn>3</mn><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>3</mn><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>8</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa91">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0355" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>3</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>21</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mo stretchy="false">(</mo><mn>3</mn><mi>y</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa92">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0356" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>−</mo><mfrac><mi>y</mi><mn>3</mn></mfrac><mo>=</mo><mo>−</mo><mn>7</mn></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mfrac><mi>x</mi><mn>3</mn></mfrac><mo>−</mo><mfrac><mi>y</mi><mn>2</mn></mfrac><mo>=</mo><mo>−</mo><mn>8</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa93">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0357" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>7</mn></mfrac><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>1</mn><mrow><mn>14</mn></mrow></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa94">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0359" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mfrac><mi>x</mi><mn>4</mn></mfrac><mo>−</mo><mfrac><mi>y</mi><mn>2</mn></mfrac><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mfrac><mi>x</mi><mn>3</mn></mfrac><mo>+</mo><mfrac><mi>y</mi><mn>6</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa95">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0360" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>10</mn></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa96">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0362" display="block"><mrow><mrow><mo>{</mo><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><mn>15</mn></mrow></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mrow><mn>12</mn></mrow></mfrac><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>3</mn><mrow><mn>10</mn></mrow></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>8</mn></mfrac><mi>y</mi><mo>=</mo><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa97">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0364" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>0.2</mn><mi>x</mi><mo>−</mo><mn>0.05</mn><mi>y</mi><mo>=</mo><mn>0.43</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>0.3</mn><mi>x</mi><mo>+</mo><mn>0.1</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>0.3</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa98">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0365" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>0.1</mn><mi>x</mi><mo>+</mo><mn>0.3</mn><mi>y</mi><mo>=</mo><mn>0.3</mn></mtd></mtr><mtr><mtd columnalign="left"><mn>0.05</mn><mi>x</mi><mo>−</mo><mn>0.5</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>0.63</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa99">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0366" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mn>0.15</mn><mi>x</mi><mo>−</mo><mn>0.25</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>0.3</mn></mtd></mtr><mtr><mtd columnalign="left"><mo>−</mo><mn>0.75</mn><mi>x</mi><mo>+</mo><mn>1.25</mn><mi>y</mi><mo>=</mo><mo>−</mo><mn>4</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa100">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0367" display="block"><mrow><mrow><mo>{</mo><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mo>−</mo><mn>0.15</mn><mi>x</mi><mo>+</mo><mn>1.25</mn><mi>y</mi><mo>=</mo><mn>0.4</mn></mtd></mtr><mtr><mtd columnalign="left"><mo>−</mo><mn>0.03</mn><mi>x</mi><mo>+</mo><mn>0.25</mn><mi>y</mi><mo>=</mo><mn>0.08</mn></mtd></mtr></mtable></mrow></mrow></math></span>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch03_s02_qs01_qd04">
<h3 class="title">Part D: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch03_s02_qs01_qd04_qd01" start="101">
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa101">
<div class="question">
<p class="para" id="fwk-redden-ch03_s02_qs01_p204">Explain to a beginning algebra student how to choose a method for solving a system of two linear equations. Also, explain what solutions look like and why.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa102">
<div class="question">
<p class="para" id="fwk-redden-ch03_s02_qs01_p205">Make up your own linear system with two variables and solve it using all three methods. Explain which method was preferable in your exercise.</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch03_s02_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s02_qs01_p03_ans">(−2, 11)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s02_qs01_p07_ans">(2, 2)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s02_qs01_p11_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0226" display="inline"><mo>Ø</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s02_qs01_p15_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0230" display="inline"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mtext> </mtext><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa08_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s02_qs01_p19_ans">(4, −2)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa10_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s02_qs01_p23_ans">(3, −2)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa12_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa13_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s02_qs01_p27_ans">(3, −4)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa14_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa15_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0241" display="block"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>8</mn><mn>5</mn></mfrac><mo>,</mo><mo>−</mo><mfrac><mn>7</mn><mrow><mn>10</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa16_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa17_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s02_qs01_p35_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0245" display="inline"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa18_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa19_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s02_qs01_p39_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0249" display="inline"><mo>Ø</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa20_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa21_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s02_qs01_p43_ans">(2, −3)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa22_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa23_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0255" display="block"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mtext> </mtext><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa24_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa25_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s02_qs01_p51_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch03_m0259" display="inline"><mo>Ø</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa26_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa27_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch03_s02_qs01_p55_ans">(1, 1)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa28_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa29_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0265" display="block"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><mn>11</mn></mrow><mrow><mn>10</mn></mrow></mfrac><mo>,</mo><mo>−</mo><mfrac><mn>2</mn><mn>5</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa30_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa31_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch03_m0268" display="block"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mtext> </mtext><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa32_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch03_s02_qs01_qa33_ans">
<div class="answer">