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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]

Sagot :

GinnyAnswer
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]
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