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]
