Question

What is the y-value in the solution to this system of linear equations?

4x + 5y = −12

-2x + 3y = −16

−4
−2
2
5

Answer 1

To find the y-value in the solution to the system of equations:

1. **Equations:**
- \(4x + 5y = -12\) (Equation 1)
- \(-2x + 3y = -16\) (Equation 2)

2. **Step 1: Solve one equation for x or y.**
Let's solve Equation 1 for \(y\):
\[
5y = -12 - 4x \\
y = \frac{-12 - 4x}{5}
\]

3. **Step 2: Substitute for y in the second equation.**
Plug \(y\) from Equation 1 into Equation 2:
\[
-2x + 3\left(\frac{-12 - 4x}{5}\right) = -16
\]
Multiply everything by 5 to eliminate the fraction:
\[
-10x + 3(-12 - 4x) = -80 \\
-10x - 36 - 12x = -80 \\
-22x - 36 = -80
\]
Add 36 to both sides:
\[
-22x = -44
\]
Divide by -22:
\[
x = 2
\]

4. **Step 3: Substitute back to find y.**
Now that we have \(x = 2\), substitute it back into Equation 1 to find \(y\):
\[
4(2) + 5y = -12 \\
8 + 5y = -12 \\
5y = -12 - 8 \\
5y = -20
\]
Divide both sides by 5:
\[
y = -4
\]

Thus, the y-value in the solution to the system of linear equations is \(\boxed{-4}\).