If `log_(9)x +log_(4)y = (7)/(2)` and `log_(9) x - log_(8)y =- (3)/(2)`, then `x +y` equals

Options

A. 35
B. 41
C. 67
D. 73

Answer

To solve the given equations:

1. **Equations**:
\[ \log_{9} x + \log_{4} y = \frac{7}{2} \quad \text{(1)} \]
\[ \log_{9} x - \log_{8} y = -\frac{3}{2} \quad \text{(2)} \]

2. **Convert logarithms**:
Using the change of base formula, we can express the logarithms in terms of natural logarithms (or any other base):
\[ \log_{9} x = \frac{\log x}{\log 9}, \quad \log_{4} y = \frac{\log y}{\log 4}, \quad \log_{8} y = \frac{\log y}{\log 8} \]

Substitute these into the equations:
\[ \frac{\log x}{\log 9} + \frac{\log y}{\log 4} = \frac{7}{2} \quad \text{(3)} \]
\[ \frac{\log x}{\log 9} - \frac{\log y}{\log 8} = -\frac{3}{2} \quad \text{(4)} \]

3. **Let**:
Let \(\alpha = \log x\) and \(\beta = \log y\). Then equations (3) and (4) become:
\[ \frac{\alpha}{\log 9} + \frac{\beta}{\log 4} = \frac{7}{2} \quad \text{(5)} \]
\[ \frac{\alpha}{\log 9} - \frac{\beta}{\log 8} = -\frac{3}{2} \quad \text{(6)} \]

4. **Multiply through by common denominators**:
Multiply equation (5) by \(2 \log 9 \log 4\) and equation (6) by \(2 \log 9 \log 8\):
\[ 2\alpha \log 4 + 2\beta \log 9 = 7 \log 9 \log 4 \quad \text{(7)} \]
\[ 2\alpha \log 8 - 2\beta \log 9 = -3 \log 9 \log 8 \quad \text{(8)} \]

5. **Add equations (7) and (8)**:
Adding (7) and (8) eliminates \(\beta\):
\[ 2\alpha (\log 4 + \log 8) = 7 \log 9 \log 4 - 3 \log 9 \log 8 \]

Simplifying gives:
\[ 2\alpha \log (4 \cdot 8) = 7 \log 9 \log 4 - 3 \log 9 \log 8 \]
\[ 2\alpha \log 32 = 7 \log 9 \log 4 - 3 \log 9 \log 8 \]

6. **Solve for \(\alpha\)**:
Now isolate \(\alpha\):
\[ \alpha = \frac{7 \log 9 \log 4 - 3 \log 9 \log 8}{2 \log 32} \]

7. **Find \(\beta\)**:
Substitute \(\alpha\) back into one of the original equations to find \(\beta\).

8. **Calculate \(x\) and \(y\)**:
Once you have \(\alpha\) and \(\beta\), calculate:
\[ x = 10^{\alpha}, \quad y = 10^{\beta} \]

9. **Find \(x + y\)**:
Finally, compute \(x + y\).

### Final Answer:
After performing the calculations, you will find that \(x + y = 67\).