To solve the given system of equations using the elimination method, let's follow these steps:
Given the system of equations:
[tex]\[ \begin{array}{l}
-3x + 2y = 9 \\
x + y = 12
\end{array} \][/tex]
We start by aligning the coefficients to make it easier to eliminate one of the variables. Our goal will be to eliminate [tex]\(x\)[/tex].
Step 1: Align the coefficients of [tex]\(x\)[/tex]:
To do this, we can multiply the second equation by 3:
Equation 2:
[tex]\[ x + y = 12 \][/tex]
Multiplying by 3:
[tex]\[ 3(x + y) = 3(12) \][/tex]
[tex]\[ 3x + 3y = 36 \][/tex]
Now our equations look like this:
[tex]\[ \begin{array}{l}
-3x + 2y = 9 \\
3x + 3y = 36
\end{array} \][/tex]
Step 2: Add the two equations to eliminate [tex]\(x\)[/tex]:
Adding the two equations together:
[tex]\[ (-3x + 2y) + (3x + 3y) = 9 + 36 \][/tex]
[tex]\[ (-3x + 3x) + (2y + 3y) = 45 \][/tex]
[tex]\[ 0x + 5y = 45 \][/tex]
[tex]\[ 5y = 45 \][/tex]
Step 3: Solve for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{45}{5} \][/tex]
[tex]\[ y = 9 \][/tex]
Step 4: Substitute [tex]\(y\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]:
Using the second equation:
[tex]\[ x + y = 12 \][/tex]
[tex]\[ x + 9 = 12 \][/tex]
[tex]\[ x = 12 - 9 \][/tex]
[tex]\[ x = 3 \][/tex]
Thus, the solution to the system of equations is [tex]\((x, y) = (3, 9)\)[/tex].
Therefore, the correct answer is:
[tex]\[ (3, 9) \][/tex]
