## Solving the System of Equations by Elimination Method

We are given the following system of equations:

**(1)** (a - b)x + (a + b)y = a² - 2ab - b²

**(2)** (a + b)(x + y) = a² + b²

Our goal is to solve for *x* and *y* using the elimination method.

### 1. Simplify Equation (2)

First, we need to simplify equation (2) to make it easier to work with:

**(2)** (a + b)x + (a + b)y = a² + b²

### 2. Align Similar Terms

Now, let's arrange both equations so that the *x* and *y* terms are aligned:

**(1)** (a - b)x + (a + b)y = a² - 2ab - b²

**(2)** (a + b)x + (a + b)y = a² + b²

### 3. Eliminate One Variable

To eliminate *y*, we can multiply equation (2) by -1:

**(1)** (a - b)x + (a + b)y = a² - 2ab - b²

**(2)** -(a + b)x - (a + b)y = -a² - b²

Now, adding the two equations together, the *y* terms cancel out:

**(1) + (2)**
(a - b)x - (a + b)x = a² - 2ab - b² - a² - b²

Simplifying, we get:

-2bx = -2ab - 2b²

Dividing both sides by -2b:

**x = a + b**

### 4. Solve for the Other Variable

Now that we know the value of *x*, we can substitute it into either of the original equations to solve for *y*. Let's use equation (1):

**(1)** (a - b)(a + b) + (a + b)y = a² - 2ab - b²

Simplifying:

a² - b² + (a + b)y = a² - 2ab - b²

Canceling out the a² and b² terms:

(a + b)y = -2ab

Dividing both sides by (a + b):

**y = -2a**

### 5. Solution

Therefore, the solution to the system of equations is:

**x = a + b****y = -2a**