Select the correct answer.

What is the solution to this system of equations?
[tex]\[
\begin{aligned}
x & = 12 - y \\
2x + 3y & = 29
\end{aligned}
\][/tex]

A. [tex]\( x = 8, y = 4 \)[/tex]

B. [tex]\( x = 6, y = 6 \)[/tex]

C. [tex]\( x = 7, y = 5 \)[/tex]

D. [tex]\( x = 9, y = 3 \)[/tex]



Answer :

To solve the given system of equations:
[tex]\[ \begin{aligned} x & = 12 - y \\ 2x + 3y & = 29 \end{aligned} \][/tex]

We start by substituting the expression for [tex]\( x \)[/tex] from the first equation into the second equation:

1. Substitute [tex]\( x = 12 - y \)[/tex] into [tex]\( 2x + 3y = 29 \)[/tex]:
[tex]\[ 2(12 - y) + 3y = 29 \][/tex]

2. Distribute the 2 inside the parentheses:
[tex]\[ 24 - 2y + 3y = 29 \][/tex]

3. Combine like terms:
[tex]\[ 24 + y = 29 \][/tex]

4. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = 29 - 24 \][/tex]
[tex]\[ y = 5 \][/tex]

With [tex]\( y = 5 \)[/tex], we substitute back into the first equation to find [tex]\( x \)[/tex]:

5. Substitute [tex]\( y = 5 \)[/tex] into [tex]\( x = 12 - y \)[/tex]:
[tex]\[ x = 12 - 5 \][/tex]
[tex]\[ x = 7 \][/tex]

So, the solution to the system of equations is [tex]\( x = 7 \)[/tex] and [tex]\( y = 5 \)[/tex].

Thus, the correct answer is:
[tex]\[ \boxed{C: \ x = 7, \ y = 5} \][/tex]