Question

Solve using elimination.

8x − 9y = –8
–9x + 9y = 18

Answer 1

To solve the system of equations using the elimination method, we start with the two equations:

1. \( 8x - 9y = -8 \) (Equation 1)
2. \( -9x + 9y = 18 \) (Equation 2)

Our goal is to eliminate one of the variables by manipulating the equations. We can do this by making the coefficients of \( y \) in both equations equal.

### Step 1: Manipulate the Equations

To eliminate \( y \), we can multiply both equations so that the coefficients of \( y \) are equal. We can multiply Equation 1 by \( 1 \) (which keeps it the same) and Equation 2 by \( 1 \) as well (since the coefficients of \( y \) are already the same in magnitude but opposite in sign). This gives us:

- Keep Equation 1 as is:
\[
8x - 9y = -8
\]

- Rewrite Equation 2 by multiplying by \( 1 \) (no change):
\[
-9x + 9y = 18
\]

### Step 2: Add the Equations

We can now add both equations to eliminate \( y \):
\[
(8x - 9y) + (-9x + 9y) = -8 + 18
\]
This simplifies to:
\[
(8x - 9x) + (-9y + 9y) = -8 + 18
\]
\[
-x + 0 = 10
\]

### Step 3: Solve for \( x \)

This simplifies to:
\[
-x = 10 \implies x = -10
\]

### Step 4: Substitute \( x \) back into one of the original equations

Now, substitute \( x = -10 \) into one of the original equations. We can use Equation 1:
\[
8(-10) - 9y = -8
\]
This simplifies to:
\[
-80 - 9y = -8
\]
Add \( 80 \) to both sides:
\[
-9y = -8 + 80
\]
\[
-9y = 72
\]
Now, divide by \( -9 \):
\[
y = \frac{72}{-9} = -8
\]

### Step 5: Write the Solution

The solution to the system of equations is:
\[
x = -10, \quad y = -8
\]

Thus, the solution is:
\[
\boxed{(-10, -8)}
\]