To solve the system of equations using the elimination method, follow these steps:
Given system of equations:
[tex]\[
\begin{array}{l}
-3x + 2y = 9 \\
x + y = 12
\end{array}
\][/tex]
### Step 1: Align the Equations
Align the two equations to prepare for the elimination method:
[tex]\[
\begin{array}{l}
-3x + 2y = 9 \\
x + y = 12
\end{array}
\][/tex]
### Step 2: Eliminate One Variable
Our goal is to eliminate one of the variables, in this case, we will eliminate [tex]\(x\)[/tex]. To do so, we can multiply the second equation by 3 to align the coefficients of [tex]\(x\)[/tex] between the two equations:
Multiply the second equation by 3:
[tex]\[
3(x + y) = 3 \cdot 12
\][/tex]
This simplifies to:
[tex]\[
3x + 3y = 36
\][/tex]
Now, we have:
[tex]\[
\begin{array}{l}
-3x + 2y = 9 \\
3x + 3y = 36
\end{array}
\][/tex]
### Step 3: Add the Equations
Add the two equations to eliminate [tex]\(x\)[/tex]:
[tex]\[
(-3x + 2y) + (3x + 3y) = 9 + 36
\][/tex]
This simplifies to:
[tex]\[
5y = 45
\][/tex]
### Step 4: Solve for [tex]\(y\)[/tex]
Solve for [tex]\(y\)[/tex] by dividing both sides of the equation by 5:
[tex]\[
y = \frac{45}{5} = 9
\][/tex]
### Step 5: Substitute [tex]\(y\)[/tex] Back
Substitute [tex]\(y = 9\)[/tex] back into the original second equation to find [tex]\(x\)[/tex]:
[tex]\[
x + y = 12
\][/tex]
[tex]\[
x + 9 = 12
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = 12 - 9 = 3
\][/tex]
### Step 6: Verify the Solution
Verify that the solution satisfies both original equations:
1. For [tex]\(-3x + 2y = 9\)[/tex]:
[tex]\[
-3(3) + 2(9) = -9 + 18 = 9
\][/tex]
2. For [tex]\(x + y = 12\)[/tex]:
[tex]\[
3 + 9 = 12
\][/tex]
Both equations are satisfied.
### Solution
The solution to the system of equations is:
[tex]\[
(x, y) = (3, 9)
\][/tex]
Therefore, the correct option is [tex]\( (3, 9) \)[/tex].
