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Answer:
What is the elimination method?
The elimination method is a technique for solving systems of linear equations. Let's walk through a couple of examples.
Example 1
We're asked to solve this system of equations:
\begin{aligned} 2y+7x &= -5\\\\ 5y-7x &= 12 \end{aligned}2y+7x5y−7x=−5=12
We notice that the first equation has a 7x7x7, x term and the second equation has a -7x−7xminus, 7, x term. These terms will cancel if we add the equations together—that is, we'll eliminate the xxx terms:
\begin{aligned} 2y+\redD{7x} &= -5 \\ +~5y\redD{-7x}&=12\\ \hline\\ 7y+0 &=7 \end{aligned}2y+7x+ 5y−7x7y+0=−5=12=7
Solving for yyy, we get:
\begin{aligned} 7y+0 &=7\\\\ 7y &=7\\\\ y &=\goldD{1} \end{aligned}7y+07yy=7=7=1
Plugging this value back into our first equation, we solve for the other variable:
\begin{aligned} 2y+7x &= -5\\\\ 2\cdot \goldD{1}+7x &= -5\\\\ 2+7x&=-5\\\\ 7x&=-7\\\\ x&=\blueD{-1} \end{aligned}2y+7x2⋅1+7x2+7x7xx=−5=−5=−5=−7=−1
The solution to the system is x=\blueD{-1}x=−1x, equals, start color #11accd, minus, 1, end color #11accd, y=\goldD{1}y=1y, equals, start color #e07d10, 1, end color #e07d10.
We can check our solution by plugging these values back into the original equations. Let's try the second equation:
\begin{aligned} 5y-7x &= 12\\\\ 5\cdot\goldD{1}-7(\blueD{-1}) &\stackrel ?= 12\\\\ 5+7 &= 12 \end{aligned}5y−7x5⋅1−7(−1)5+7=12=?12=12
Yes, the solution checks out.
If you feel uncertain why this process works, check out this intro video for an in-depth walkthrough.
Example 2
We're asked to solve this system of equations:
\begin{aligned} -9y+4x - 20&=0\\\\ -7y+16x-80&=0 \end{aligned}−9y+4x−20−7y+
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