` ( 10 )/( x+ y) + (2 ) /(x - y ) = 4`,
` ( 15)/( x + y ) - (9) /( x - y) = - 2 `.

Options

A. ` x = ( 21)/(8), y = (9)/(8)`
B. ` x = ( 21)/(4), y = (9)/(8)`
C. ` x = ( 21)/(8), y = -(9)/(8)`
D. ` x = ( 21)/(8), y = (9)/(4)`

Answer

To solve the given system of equations:

1. **Equations**:
\[ \frac{10}{x+y} + \frac{2}{x-y} = 4 \quad \text{(1)} \]
\[ \frac{15}{x+y} - \frac{9}{x-y} = -2 \quad \text{(2)} \]

2. **Substitution**:
Let:
\[ a = \frac{1}{x+y} \quad \text{and} \quad b = \frac{1}{x-y} \]
Then, we can rewrite the equations as:
\[ 10a + 2b = 4 \quad \text{(3)} \]
\[ 15a - 9b = -2 \quad \text{(4)} \]

3. **Multiply to Eliminate \(b\)**:
To eliminate \(b\), we can multiply equation (3) by 9 and equation (4) by 2:
\[ 90a + 18b = 36 \quad \text{(5)} \]
\[ 30a - 18b = -4 \quad \text{(6)} \]

4. **Add Equations (5) and (6)**:
Adding equations (5) and (6):
\[ 90a + 18b + 30a - 18b = 36 - 4 \]
This simplifies to:
\[ 120a = 32 \]

5. **Solve for \(a\)**:
Dividing both sides by 120:
\[ a = \frac{32}{120} = \frac{4}{15} \]

6. **Substitute \(a\) back into Equation (3)**:
Substitute \(a\) into equation (3):
\[ 10\left(\frac{4}{15}\right) + 2b = 4 \]
This simplifies to:
\[ \frac{40}{15} + 2b = 4 \]
\[ 2b = 4 - \frac{40}{15} \]
Converting 4 to a fraction with a denominator of 15:
\[ 2b = \frac{60}{15} - \frac{40}{15} = \frac{20}{15} \]
Thus:
\[ b = \frac{10}{15} = \frac{2}{3} \]

7. **Find \(x+y\) and \(x-y\)**:
From our substitutions:
\[ \frac{1}{x+y} = a = \frac{4}{15} \implies x+y = \frac{15}{4} \]
\[ \frac{1}{x-y} = b = \frac{2}{3} \implies x-y = \frac{3}{2} \]

8. **Set Up the System of Equations**:
Now we have:
\[ x+y = \frac{15}{4} \quad \text{(7)} \]
\[ x-y = \frac{3}{2} \quad \text{(8)} \]

9. **Add Equations (7) and (8)**:
Adding equations (7) and (8):
\[ (x+y) + (x-y) = \frac{15}{4} + \frac{3}{2} \]
Converting \(\frac{3}{2}\) to a fraction with a denominator of 4:
\[ \frac{3}{2} = \frac{6}{4} \]
Thus:
\[ 2x = \frac{15}{4} + \frac{6}{4} = \frac{21}{4} \]
Therefore:
\[ x = \frac{21}{8} \]

10. **Substitute \(x\) back to find \(y\)**:
Substitute \(x\) into equation (7):
\[ \frac{21}{8} + y = \frac{15}{4} \]
Converting \(\frac{15}{4}\) to a fraction with a denominator of 8:
\[ \frac{15}{4} = \frac{30}{8} \]
Thus:
\[ y = \frac{30}{8} - \frac{21}{8} = \frac{9}{8} \]

11. **Final Values**:
Therefore, the solution is:
\[ x = \frac{21}{8}, \quad y = \frac{9}{8} \]