Question

Solve using elimination.

–7x − 9y = 9
8x + 7y = 16

Answer 1

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

1) \(-7x - 9y = 9\)
2) \(8x + 7y = 16\)

We will eliminate one of the variables by manipulating both equations so they can help us add or subtract them.

First, let's multiply Equation (1) by \(8\) and Equation (2) by \(7\). This will allow us to align the coefficients of \(x\):

\[
8(-7x - 9y) = 8(9) \implies -56x - 72y = 72 \quad \text{(Equation 3)}
\]

\[
7(8x + 7y) = 7(16) \implies 56x + 49y = 112 \quad \text{(Equation 4)}
\]

Now, we have:

3) \(-56x - 72y = 72\)
4) \(56x + 49y = 112\)

Next, we will add Equations (3) and (4):

\[
(-56x - 72y) + (56x + 49y) = 72 + 112
\]

This simplifies to:

\[
(-72y + 49y) = 184
\]

Combine the terms:

\[
-23y = 184
\]

Now, solve for \(y\):

\[
y = \frac{184}{-23} = -8
\]

Now that we have the value of \(y\), we can substitute it back into one of the original equations to solve for \(x\). Let's use Equation (2):

\[
8x + 7(-8) = 16
\]

This simplifies to:

\[
8x - 56 = 16
\]

Now, add \(56\) to both sides:

\[
8x = 72
\]

Divide by \(8\):

\[
x = 9
\]

Thus, the solution to the system of equations is:

\[
x = 9, \quad y = -8
\]

So, the solution in ordered pair form is:

\[
(9, -8)
\]