Question

Use the elimination method to solve the follow system of equations. 9x−7y=22 x+3y=−24 This question requires you to show your work. (1 point) Responses (−3,−7) open paren negative 3 comma negative 7 close paren (3,−9) open paren 3 comma negative 9 close paren (4, 2) (4, 2) (0,−8)

Answer 1

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

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

Next, we aim to eliminate one variable by making the coefficients of \( x \) or \( y \) the same in both equations.

**Step 1: Manipulate Equation 2 to align with Equation 1.**

We will multiply Equation 2 by 9 so that the coefficients of \( x \) will match:

\[
9(x + 3y) = 9(-24)
\]
This simplifies to:

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

**Step 2: Rewrite the equations.**

Now we have:
1. \( 9x - 7y = 22 \)
2. \( 9x + 27y = -216 \)

**Step 3: Eliminate \( x \) by subtracting Equation 1 from Equation 3.**

Subtract Equation 1 from Equation 3:

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

This will result in:

\[
9x + 27y - 9x + 7y = -238
\]
Combining the \( y \) terms yields:

\[
34y = -238
\]

**Step 4: Solve for \( y \).**

To isolate \( y \):

\[
y = \frac{-238}{34}
\]
Simplifying:

\[
y = -7
\]

**Step 5: Substitute \( y \) back to solve for \( x \).**

Now that we have \( y = -7 \), we can substitute this value back into Equation 2 to find \( x \):

\[
x + 3(-7) = -24
\]
This simplifies to:

\[
x - 21 = -24
\]

Adding 21 to both sides gives:

\[
x = -24 + 21
\]
\[
x = -3
\]

**Step 6: Write the solution as an ordered pair.**

The solution to the system of equations is:

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

Thus, the answer is \( (-3, -7) \).

**Final Answer:**
\((-3, -7)\)