Find the value of x and y:` ax +by =a-b `
` bx-ay =a+b`

Answer

To solve the equations \( ax + by = a - b \) and \( bx - ay = a + b \) for \( x \) and \( y \), we will use the elimination method. Here’s a step-by-step solution:

### Step 1: Write down the equations
We have the following two equations:
1. \( ax + by = a - b \) (Equation 1)
2. \( bx - ay = a + b \) (Equation 2)

### Step 2: Make the coefficients of \( x \) the same
To eliminate \( x \), we need to make the coefficients of \( x \) in both equations the same. We can do this by multiplying Equation 1 by \( b \) and Equation 2 by \( a \).

- Multiply Equation 1 by \( b \):
\[ b(ax + by) = b(a - b) \implies abx + b^2y = ab - b^2 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by \( a \):
\[ a(bx - ay) = a(a + b) \implies abx - a^2y = a^2 + ab \quad \text{(Equation 4)} \]

### Step 3: Subtract the equations to eliminate \( x \)
Now, we will subtract Equation 4 from Equation 3 to eliminate \( x \):
\[ (abx + b^2y) - (abx - a^2y) = (ab - b^2) - (a^2 + ab) \]
This simplifies to:
\[ b^2y + a^2y = ab - b^2 - a^2 - ab \]
\[ (b^2 + a^2)y = -b^2 - a^2 \]

### Step 4: Solve for \( y \)
Now, we can isolate \( y \):
\[ y = \frac{-b^2 - a^2}{b^2 + a^2} \]
This simplifies to:
\[ y = -1 \]

### Step 5: Substitute \( y \) back to find \( x \)
Now that we have \( y = -1 \), we can substitute this value back into one of the original equations to find \( x \). We will use Equation 1:
\[ ax + b(-1) = a - b \]
This simplifies to:
\[ ax - b = a - b \]
Adding \( b \) to both sides gives:
\[ ax = a \]
Dividing both sides by \( a \) (assuming \( a \neq 0 \)):
\[ x = 1 \]

### Final Answer
Thus, the values of \( x \) and \( y \) are:
\[ x = 1, \quad y = -1 \]