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How do you solve the system $3x+2y=-17$ and $2x-5y=14$ by substitution?

seo-qna
Last updated date: 29th Feb 2024
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IVSAT 2024
Answer
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Hint: From the question given, we have been asked to solve the system \[3x+2y=-17,2x-5y=14\]. We can solve the given question by understanding the question correctly and applying some basic transformations to the given system of equations. Then we have to simplify further to get the given system of equations solved.

Complete step-by-step solution:
From the question given, we have been given a system of equations \[3x+2y=-17,2x-5y=14\].
We can clearly observe that, in the given system of equations, the left hand side of both the equations are not equal to \[y\]. So we need to choose any one equation and equate its left hand side to $y$ .
After doing that for \[2x-5y=14\] we will have
$\begin{align}
  & \Rightarrow -5y=14-2x \\
 & \Rightarrow 5y=2x-14 \\
 & \Rightarrow y=\dfrac{2x-14}{5} \\
\end{align}$
Now we need to substitute this value of $y$ in the other equation. After doing that we will have
\[\begin{align}
  & 3x+2\left( \dfrac{2x-14}{5} \right)=-17 \\
 & \Rightarrow 3x+\left( \dfrac{4x-28}{5} \right)+17=0 \\
 & \Rightarrow \dfrac{15x+4x-28+85}{5}=0 \\
 & \Rightarrow \dfrac{19x+57}{5}=0 \\
 & \Rightarrow 19x=-57 \\
 & \Rightarrow x=-3 \\
\end{align}\]
Now, substitute for \[x\] in any of those two equations to get the value of \[y\].
Therefore,
\[\begin{align}
  & \Rightarrow y=\left( \dfrac{2\left( -3 \right)-14}{5} \right) \\
 & \Rightarrow y=\left( \dfrac{-6-14}{5} \right) \\
 & \Rightarrow y=\dfrac{-20}{5} \\
 & \Rightarrow y=-4 \\
\end{align}\]
Hence, \[x=-3,y=-4\]
Hence, the given systems of equations are solved.

Note: We should be well aware of solving the given system of equations using some basic transformations to the given question. After getting the value of one variable, we should use it and get the value of another variable. Also, we should be very careful while doing the calculation part. It will be efficient if we verify the answer before finalizing it. In this case verification is done by substituting the value of $x$ in both the equations and observing if the value of $y$ is the same. For \[2x-5y=14\] if we substitute $x=-3$ we have $y=-4$ we need to check for \[3x+2y=-17\] if we substitute $x=-3$ we have
$\begin{align}
  & \Rightarrow 3\left( -3 \right)+2y=-17 \\
 & \Rightarrow -9+2y=-17 \\
 & \Rightarrow 2y=-17+9 \\
 & \Rightarrow 2y=-8 \\
 & \Rightarrow y=-4 \\
\end{align}$



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