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Write an equivalent system with integer coefficients.

[tex]\[
\begin{array}{l}
\frac{2}{3} x + \frac{3}{5} y = \frac{172}{15} \\
\frac{7}{8} x + 3 y = 77
\end{array}
\][/tex]

A. [tex]\(10 x + 9 y = 1\)[/tex]

B. [tex]\(9 x + 10 y = 172\)[/tex]

C. [tex]\(10 x + 9 y = 172\)[/tex]

D. [tex]\(7 x + 24 y = 616\)[/tex]



Answer :

GinnyAnswer
Let's transform the given system of equations with fractions into a system with integer coefficients.

Here are the equations provided:
[tex]\[ \frac{2}{3}x + \frac{3}{5}y = \frac{172}{15} \][/tex]
[tex]\[ \frac{7}{8}x + 3y = 77 \][/tex]

### Step-by-Step Solution:

1. Clear the Fractions in the First Equation:
[tex]\[ \frac{2}{3}x + \frac{3}{5}y = \frac{172}{15} \][/tex]

Find the least common multiple (LCM) of the denominators 3, 5, and 15, which is 15.

Multiply both sides of the equation by 15:
[tex]\[ 15 \left(\frac{2}{3}x\right) + 15 \left(\frac{3}{5}y\right) = 15 \left(\frac{172}{15}\right) \][/tex]

Simplify:
[tex]\[ 5 \cdot 2x + 3 \cdot 3y = 172 \][/tex]
[tex]\[ 10x + 9y = 172 \][/tex]

2. Clear the Fractions in the Second Equation:
[tex]\[ \frac{7}{8}x + 3y = 77 \][/tex]

Multiply both sides of the equation by 8 to eliminate the fraction:
[tex]\[ 8 \left(\frac{7}{8}x\right) + 8 \cdot 3y = 8 \cdot 77 \][/tex]

Simplify:
[tex]\[ 7x + 24y = 616 \][/tex]

The resulting system of equations with integer coefficients is:
[tex]\[ 10x + 9y = 172 \][/tex]
[tex]\[ 7x + 24y = 616 \][/tex]

### Conclusion:
Therefore, the correct answer is:
[tex]\[ \text{d.} \ 10x + 9y = 172 \][/tex]
[tex]\[ \phantom{\text{d.} \ } 7x + 24y = 616 \][/tex]

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