Select the correct answer.

What is the solution of the system [tex] x - 3y = -13 [/tex] and [tex] 5x + 7y = 34 [/tex]?

A. [tex] x = \frac{6}{5}, y = 4 [/tex]
B. [tex] x = \frac{7}{2}, y = \frac{9}{2} [/tex]
C. [tex] x = \frac{1}{2}, y = \frac{9}{2} [/tex]
D. [tex] x = -3, y = 7 [/tex]



Answer :

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].