Question

use the elimination method to solve the following system of equations.
9x-7y=22
x+3y=-24

Answer 1

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

1. \( 9x - 7y = 22 \) (Equation 1)
2. \( x + 3y = -24 \) (Equation 2)

To eliminate one variable, we can manipulate Equation 2 to have the same coefficient for \(x\) as in Equation 1. First, we can multiply Equation 2 by \(9\):

\[
9(x + 3y) = 9(-24)
\]

This results in:

\[
9x + 27y = -216 \quad \text{(Equation 3)}
\]

Now, we have the following system of equations:

1. \( 9x - 7y = 22 \) (Equation 1)
2. \( 9x + 27y = -216 \) (Equation 3)

Next, we will subtract Equation 1 from Equation 3 to eliminate \(x\):

\[
(9x + 27y) - (9x - 7y) = -216 - 22
\]

This simplifies to:

\[
9x + 27y - 9x + 7y = -238
\]

Thus, we get:

\[
34y = -238
\]

Now we can solve for \(y\):

\[
y = \frac{-238}{34} = -7
\]

Now that we have \(y\), we can substitute this value back into Equation 2 to solve for \(x\):

\[
x + 3(-7) = -24
\]

This simplifies to:

\[
x - 21 = -24
\]

Adding \(21\) to both sides gives:

\[
x = -3
\]

Therefore, the solution to the system of equations is:

\[
x = -3, \quad y = -7
\]

We can write the final answer as:

\[
(x, y) = (-3, -7)
\]