To solve the given system of linear equations using the elimination method, we first rewrite the system of equations clearly:
[tex]\[
\begin{cases}
4y + x = -17 \\
-2y + 2x = 16
\end{cases}
\][/tex]
Step 1: Eliminate one of the variables
We can start by eliminating [tex]\(x\)[/tex]. To do this, we need to manipulate the equations so that adding or subtracting them will result in one of the variables canceling out.
Given our equations:
1. [tex]\( 4y + x = -17 \)[/tex]
2. [tex]\(-2y + 2x = 16 \)[/tex]
Let's multiply the first equation by 2 to align the coefficients of [tex]\(x\)[/tex]:
[tex]\[
2(4y + x) = 2(-17)
\][/tex]
Which gives us:
[tex]\[
8y + 2x = -34
\][/tex]
Now, our system of equations is:
[tex]\[
\begin{cases}
8y + 2x = -34 \\
-2y + 2x = 16
\end{cases}
\][/tex]
Next, we subtract the second equation from the first to eliminate [tex]\(x\)[/tex]:
[tex]\[
(8y + 2x) - (-2y + 2x) = -34 - 16
\][/tex]
Simplifying this, we get:
[tex]\[
8y + 2x + 2y - 2x = -34 - 16
\][/tex]
[tex]\[
10y = -50
\][/tex]
Now, divide both sides by 10 to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-50}{10} = -5
\][/tex]
Step 2: Substitute [tex]\(y\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]
Let's substitute [tex]\(y = -5\)[/tex] into the first equation:
[tex]\[
4(-5) + x = -17
\][/tex]
Simplify and solve for [tex]\(x\)[/tex]:
[tex]\[
-20 + x = -17
\][/tex]
Add 20 to both sides:
[tex]\[
x = -17 + 20
\][/tex]
[tex]\[
x = 3
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
(x, y) = (3, -5)
\][/tex]
Let's write the solution in the form given in the problem:
[tex]\[
\boxed{(-5, 3)}
\][/tex]
So, the solution to the system [tex]\( \begin{cases} 4y + x = -17 \\ -2y + 2x = 16 \end{cases} \)[/tex] is [tex]\( x = 3 \)[/tex] and [tex]\( y = -5 \)[/tex].
