Solve for x and y, px + qy = 1 and `qx + py = ((p + q)^(2))/(p^2 + q^2)-1`.

Answer

To solve the equations \( px + qy = 1 \) and \( qx + py = \frac{(p + q)^2}{p^2 + q^2} - 1 \), we will follow these steps:

### Step 1: Express \( x \) in terms of \( y \)
From the first equation \( px + qy = 1 \), we can isolate \( x \):

\[ px = 1 - qy \]

\[ x = \frac{1 - qy}{p} \]

### Step 2: Substitute \( x \) into the second equation
Now, we substitute the expression for \( x \) into the second equation \( qx + py = \frac{(p + q)^2}{p^2 + q^2} - 1 \):

\[ q\left(\frac{1 - qy}{p}\right) + py = \frac{(p + q)^2}{p^2 + q^2} - 1 \]

### Step 3: Simplify the left side
Distributing \( q \) gives us:

\[ \frac{q(1 - qy)}{p} + py = \frac{(p + q)^2}{p^2 + q^2} - 1 \]

This simplifies to:

\[ \frac{q}{p} - \frac{q^2y}{p} + py = \frac{(p + q)^2}{p^2 + q^2} - 1 \]

### Step 4: Combine terms on the left side
Now, we combine the terms on the left side:

\[ \left(-\frac{q^2}{p} + p\right)y + \frac{q}{p} = \frac{(p + q)^2}{p^2 + q^2} - 1 \]

### Step 5: Find a common denominator for the right side
To simplify the right side, we need to find a common denominator:

\[ \frac{(p + q)^2 - (p^2 + q^2)}{p^2 + q^2} \]

### Step 6: Simplify the right side
Calculating \( (p + q)^2 - (p^2 + q^2) \):

\[ (p + q)^2 = p^2 + 2pq + q^2 \]

So,

\[ (p + q)^2 - (p^2 + q^2) = 2pq \]

Thus, the right side becomes:

\[ \frac{2pq}{p^2 + q^2} \]

### Step 7: Set up the equation
Now we have:

\[ \left(-\frac{q^2}{p} + p\right)y + \frac{q}{p} = \frac{2pq}{p^2 + q^2} \]

### Step 8: Isolate \( y \)
Now we can isolate \( y \):

\[ \left(-\frac{q^2}{p} + p\right)y = \frac{2pq}{p^2 + q^2} - \frac{q}{p} \]

### Step 9: Find a common denominator for the left side
Taking the common denominator for the left side gives:

\[ \left(-\frac{q^2}{p} + p\right)y = \frac{2pq - q(p^2 + q^2)}{p(p^2 + q^2)} \]

### Step 10: Solve for \( y \)
Now we can solve for \( y \):

\[ y = \frac{2pq - q(p^2 + q^2)}{p(p^2 + q^2)(-\frac{q^2}{p} + p)} \]

### Step 11: Simplify \( y \)
Factoring out \( q \) gives:

\[ y = \frac{q(p^2 - q^2)}{p^2 + q^2} \]

### Step 12: Substitute \( y \) back to find \( x \)
Now substitute \( y \) back into the equation for \( x \):

\[ x = \frac{1 - q\left(\frac{q(p^2 - q^2)}{p^2 + q^2}\right)}{p} \]

### Step 13: Simplify \( x \)
This simplifies to:

\[ x = \frac{p^2}{p(p^2 + q^2)} \]

### Final Answers
Thus, the solutions are:

\[ x = \frac{p}{p^2 + q^2}, \quad y = \frac{q}{p^2 + q^2} \]