Question

Solve using elimination.

–4x + 9y = –16
x − 6y = 19

Answer 1

To solve the system of equations using the elimination method:

1. The given equations are:
\[
-4x + 9y = -16 \quad \text{(Equation 1)}
\]
\[
x - 6y = 19 \quad \text{(Equation 2)}
\]

2. We will eliminate one variable. First, let's manipulate Equation 2 to align with Equation 1 in terms of the \( x \) coefficient. We can do this by multiplying Equation 2 by 4:
\[
4(x - 6y) = 4(19)
\]
This results in:
\[
4x - 24y = 76 \quad \text{(Equation 3)}
\]

3. Now, we have:
\[
-4x + 9y = -16 \quad \text{(Equation 1)}
\]
\[
4x - 24y = 76 \quad \text{(Equation 3)}
\]

4. Add Equation 1 and Equation 3 together to eliminate \( x \):
\[
(-4x + 9y) + (4x - 24y) = -16 + 76
\]
This simplifies to:
\[
0x - 15y = 60
\]
or:
\[
-15y = 60
\]

5. Solve for \( y \):
\[
y = \frac{60}{-15} = -4
\]

6. Substitute \( y = -4 \) back into one of the original equations to find \( x \). We will use Equation 2:
\[
x - 6(-4) = 19
\]
This simplifies to:
\[
x + 24 = 19
\]
Solving for \( x \):
\[
x = 19 - 24 = -5
\]

7. The solution to the system of equations is:
\[
\boxed{(-5, -4)}
\]