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<h4>Classifying and Rewriting Systems of Equations</h4>
<p>Find a solution to the system of equations using the addition method. </p>
<p>\( x + 3y = 2 \)<br>
\( 3x + 9y = 6 \)</p>
<p><strong>Solution </strong></p>
<p>With the addition method, we want to eliminate one of the variables by adding the equations. In this case, let’s focus on eliminating \( x \). If we multiply both sides of the first equation by \( -3 \), then we will be able to eliminate the \( x \)-variable. </p>
<p>Now add the equations:</p>
<p>\(\begin{array}{rcl} x + 3y &=& 2 \\ (-3)(x+3y) &=& (-3)(2) \\ -3x - 9y &=& -6\end{array} \)<br>
</p>
<p>\(\begin{array}{rcl} -3x - 9y &=& -6 \\ + 3x + 9y &=& 6 \\ 0 &=& 0\end{array} \)<br>
</p>
<p>We can see that there will be an infinite number of situations that satisfy both equations.</p>
<h4>Try It: Classifying and Rewriting Systems of Equations</h4>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="5e059590-64df-4467-974a-5452d8ee4159" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<p>Change a single constant term in the system from above so that there are no solutions.</p>
<p> \(\begin{array}{rcl} x + 3y &=& 2 \\ 3x + 9y &=& 6\end{array} \)</p>
<p>Show that you were successful in changing the system to have no solution.</p>
</div>
<div class="os-raise-ib-input-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer: Here is how to rewrite the system so there is no solution. </p>
<p>We can simply change the 2 in the first equation to any other number, such as 5.</p>
<p> \(\begin{array}{rcl} x + 3y &=& 5 \\ 3x + 9y &=& 6\end{array} \)</p>
<p>Let’s try to prove there is no solution. Using substitution:</p>
<p>\( \begin{array}{rcl}x + 3y &=& 5 \\ x &=& 5 - 3y\end{array} \)</p>
<p>Substitute this expression into the second equation.</p>
<p>\( \begin{array}{rcl}3x + 9y &=& 6 \\ 3(5 - 3y) + 9y &=& 6 \\ 15 - 9y + 9y &=& 6 \\ 15 &\neq& 6\end{array} \)</p>
<p>Since the equation is not true, the system has no solution.</p>
</div>
</div>