To solve the system of equations given by:
[tex]\[ x - 3y = -13 \][/tex]
[tex]\[ 5x + 7y = 34 \][/tex]
follow these steps:
1. Isolate one of the variables in one of the equations. Let's start with the first equation:
[tex]\[ x - 3y = -13 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 3y - 13 \][/tex]
2. Substitute this expression for [tex]\( x \)[/tex] in the second equation:
[tex]\[ 5(3y - 13) + 7y = 34 \][/tex]
3. Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[
15y - 65 + 7y = 34
\][/tex]
[tex]\[
22y - 65 = 34
\][/tex]
[tex]\[
22y = 99
\][/tex]
[tex]\[
y = \frac{99}{22}
\][/tex]
[tex]\[
y = \frac{9}{2}
\][/tex]
4. Substitute [tex]\( y = \frac{9}{2} \)[/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]
Thus, the solution to the system is:
[tex]\[ x = \frac{1}{2}, y = \frac{9}{2} \][/tex]
The correct answer is:
C. [tex]\( x = \frac{1}{2}, y = \frac{9}{2} \)[/tex]
