What will be solution of the equations, ax+by=a-b and bx-ay=a+b?

Options

A. x=1, y=2
B. x=2, y=-1
C. x= -2, y= -2
D. x=1, y=-1

Answer

To solve the equations \( ax + by = a - b \) and \( bx - ay = a + b \), we will follow these steps:

### 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: Multiply the equations
We will multiply Equation 1 by \( b \) and Equation 2 by \( a \):
- Multiplying Equation 1 by \( b \):
\[ b(ax + by) = b(a - b) \implies abx + b^2y = ab - b^2 \quad \text{(Equation 3)} \]

- Multiplying 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 two new equations
Now, we will subtract Equation 4 from Equation 3:
\[ (abx + b^2y) - (abx - a^2y) = (ab - b^2) - (a^2 + ab) \]
This simplifies to:
\[ b^2y + a^2y = -b^2 - a^2 \]
Factoring out \( y \):
\[ (b^2 + a^2)y = - (b^2 + a^2) \]

### Step 4: Solve for \( y \)
Assuming \( b^2 + a^2 \neq 0 \), we can divide both sides by \( b^2 + a^2 \):
\[ y = -1 \]

### Step 5: Substitute \( y \) back into one of the original equations
Now we substitute \( y = -1 \) back into Equation 1:
\[ ax + b(-1) = a - b \]
This simplifies to:
\[ ax - b = a - b \]
Adding \( b \) to both sides:
\[ ax = a \]
Dividing both sides by \( a \) (assuming \( a \neq 0 \)):
\[ x = 1 \]

### Final Solution
Thus, the solution to the equations is:
\[ x = 1, \quad y = -1 \]