To solve the system of linear equations [tex]\( x - 3y = -13 \)[/tex] and [tex]\( 5x + 7y = 34 \)[/tex], follow these steps:
1. Write the system of equations:
[tex]\[
\begin{cases}
x - 3y = -13 \\
5x + 7y = 34
\end{cases}
\][/tex]
2. Solve the first equation for [tex]\( x \)[/tex]:
[tex]\[
x = 3y - 13
\][/tex]
3. Substitute this expression for [tex]\( x \)[/tex] into the second equation:
[tex]\[
5(3y - 13) + 7y = 34
\][/tex]
4. Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[
15y - 65 + 7y = 34
\][/tex]
[tex]\[
22y - 65 = 34
\][/tex]
[tex]\[
22y = 99
\][/tex]
[tex]\[
y = \frac{99}{22} = \frac{9}{2}
\][/tex]
5. Substitute [tex]\( y \)[/tex] back into the expression for [tex]\( x \)[/tex]:
[tex]\[
x = 3\left(\frac{9}{2}\right) - 13
\][/tex]
[tex]\[
x = \frac{27}{2} - 13
\][/tex]
[tex]\[
x = \frac{27}{2} - \frac{26}{2}
\][/tex]
[tex]\[
x = \frac{1}{2}
\][/tex]
The solution to the system of equations is:
[tex]\[
x = \frac{1}{2}, \quad y = \frac{9}{2}
\][/tex]
Thus, the correct answer is C. [tex]\( x = \frac{1}{2}, y = \frac{9}{2} \)[/tex].
