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.
(3,−9) open paren 3 comma negative 9 close paren (4, 2) (4, 2) (0,−8) open paren 0 comma negative 8 close paren (−3,−7)

Answers

Answered by GPT-4o mini
To solve the system of equations using the elimination method, we'll follow these steps:

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

### Step 1: Align the equations
We have:
\[ 9x - 7y = 22 \quad \text{(1)} \]
\[ x + 3y = -24 \quad \text{(2)} \]

### Step 2: Eliminate \(x\) or \(y\)
To eliminate \(x\), we can multiply Equation 2 by 9 so that the coefficient of \(x\) in both equations matches.

Multiply Equation 2 by 9:
\[ 9(x + 3y) = 9(-24) \]
This simplifies to:
\[ 9x + 27y = -216 \quad \text{(Equation 3)} \]

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

### Step 3: Subtract the two equations
Now we can eliminate \(x\) by subtracting Equation 1 from Equation 3:
\[
(9x + 27y) - (9x - 7y) = -216 - 22
\]
This simplifies to:
\[
27y + 7y = -216 - 22
\]
\[
34y = -238
\]
Now, divide both sides by 34:
\[
y = -\frac{238}{34} = -7
\]

### Step 4: Substitute \(y\) back into one of the original equations
Now we can substitute \(y = -7\) back into Equation 2 to find \(x\):
\[
x + 3(-7) = -24
\]
\[
x - 21 = -24
\]
Now add 21 to both sides:
\[
x = -24 + 21
\]
\[
x = -3
\]

### Step 5: Conclusion
The solution to the system of equations is:
\[
(x, y) = (-3, -7)
\]

Thus, the final answer is:
\[
\boxed{(-3, -7)}
\]
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