If `(1)/(x+y) + (1)/(x-y) = 4` and `(2)/(x+y) + (3)/(x-y) = 9` what is the value of x ?

Options

A. `(2)/(3)`
B. `1`
C. `(3)/(2)`
D. 2

Answer

To solve the equations given in the problem, we will follow these steps:

### Step 1: Write down the equations
We have two equations:
1. \(\frac{1}{x+y} + \frac{1}{x-y} = 4\)
2. \(\frac{2}{x+y} + \frac{3}{x-y} = 9\)

### Step 2: Find a common denominator for the first equation
The common denominator for the first equation is \((x+y)(x-y)\). Thus, we can rewrite the equation as:
\[ \frac{(x-y) + (x+y)}{(x+y)(x-y)} = 4 \]
This simplifies to:
\[ \frac{2x}{(x+y)(x-y)} = 4 \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 2x = 4(x+y)(x-y) \]

### Step 4: Expand the right-hand side
Expanding the right-hand side:
\[ 2x = 4(x^2 - y^2) \]
This simplifies to:
\[ 2x = 4x^2 - 4y^2 \]

### Step 5: Rearranging the first equation
Rearranging gives:
\[ 4x^2 - 2x - 4y^2 = 0 \quad \text{(Equation 1)} \]

### Step 6: Find a common denominator for the second equation
For the second equation, the common denominator is also \((x+y)(x-y)\):
\[ \frac{2(x-y) + 3(x+y)}{(x+y)(x-y)} = 9 \]
This simplifies to:
\[ \frac{2x - 2y + 3x + 3y}{(x+y)(x-y)} = 9 \]
Combining like terms gives:
\[ \frac{5x + y}{(x+y)(x-y)} = 9 \]

### Step 7: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 5x + y = 9(x+y)(x-y) \]

### Step 8: Expand the right-hand side
Expanding the right-hand side:
\[ 5x + y = 9(x^2 - y^2) \]
This simplifies to:
\[ 9x^2 - 5x - 9y^2 - y = 0 \quad \text{(Equation 2)} \]

### Step 9: Solve the two equations
Now we have two equations:
1. \(4x^2 - 2x - 4y^2 = 0\)
2. \(9x^2 - 5x - 9y^2 - y = 0\)

From Equation 1, we can express \(y^2\) in terms of \(x\):
\[ y^2 = x^2 - \frac{x}{2} \]

### Step 10: Substitute \(y^2\) into Equation 2
Substituting \(y^2\) into Equation 2:
\[ 9x^2 - 5x - 9\left(x^2 - \frac{x}{2}\right) - y = 0 \]

### Step 11: Solve for \(y\)
After substituting and simplifying, we can find a relationship between \(x\) and \(y\).

### Step 12: Substitute back to find \(x\)
Substituting \(y = -\frac{x}{2}\) back into either equation will allow us to solve for \(x\).

### Step 13: Final calculation
After substituting and simplifying, we find:
\[ x = \frac{2}{3} \]

### Conclusion
The value of \(x\) is \(\frac{2}{3}\).

---