Sure! Let's solve the given system of linear equations using the elimination method.
The given system is:
1. [tex]\( 4y + x = -17 \)[/tex]
2. [tex]\( -2y + 2x = 16 \)[/tex]
### Step 1: Align the coefficients of [tex]\(x\)[/tex]
To align the coefficients, we can multiply the first equation by 2 so that the coefficients of [tex]\(x\)[/tex] become the same in both equations.
Multiplying the first equation by 2:
[tex]\[ 2(4y + x) = 2(-17) \][/tex]
This simplifies to:
[tex]\[ 8y + 2x = -34 \][/tex]
Now our system of equations is:
1. [tex]\( 8y + 2x = -34 \)[/tex]
2. [tex]\( -2y + 2x = 16 \)[/tex]
### Step 2: Eliminate [tex]\(x\)[/tex]
We will eliminate [tex]\(x\)[/tex] by subtracting the second equation from the first:
[tex]\[ (8y + 2x) - (-2y + 2x) = -34 - 16 \][/tex]
This simplifies to:
[tex]\[ 8y + 2x + 2y - 2x = -34 - 16 \][/tex]
[tex]\[ 10y = -50 \][/tex]
### Step 3: Solve for [tex]\(y\)[/tex]
Now, divide both sides of the equation by 10 to solve for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{-50}{10} \][/tex]
[tex]\[ y = -5 \][/tex]
### Step 4: Substitute [tex]\(y\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]
Now that we have [tex]\(y = -5\)[/tex], we substitute this back into the first equation to solve for [tex]\(x\)[/tex]:
[tex]\[ 4(-5) + x = -17 \][/tex]
[tex]\[ -20 + x = -17 \][/tex]
Add 20 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[ x = -17 + 20 \][/tex]
[tex]\[ x = 3 \][/tex]
### Conclusion
The solution to the system of equations is:
[tex]\[ y = -5 \][/tex]
[tex]\[ x = 3 \][/tex]
So the system [tex]\( \begin{aligned} 4 y + x & = -17 \\ -2 y + 2 x & = 16 \end{aligned} \)[/tex] has the solution [tex]\( y = -5 \)[/tex] and [tex]\( x = 3 \)[/tex].
