## Solving a System of Equations using Elimination Method

This article will demonstrate how to solve the system of linear equations:

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

using the **elimination method**.

### 1. Identify the Variable to Eliminate

Observe that the coefficients of *y* are the same in both equations. Therefore, we can eliminate *y* by subtracting the equations.

### 2. Subtract the Equations

Subtracting equation (2) from equation (1) gives us:

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

Simplifying the equation:

-2bx = -2ab - 2b²

### 3. Solve for the Remaining Variable

Divide both sides of the equation by -2b (assuming b ≠ 0):

x = (2ab + 2b²) / 2b

Simplifying further:

**x = a + b**

### 4. Substitute the Value of x to Find y

Substitute the value of x (a + b) into either equation (1) or (2) to solve for *y*. Let's use equation (1):

(a-b)(a+b) + (a+b)y = a² - 2ab - b²

Expanding the left side:

a² - b² + (a+b)y = a² - 2ab - b²

Simplifying and solving for y:

(a+b)y = -2ab

**y = -2ab / (a+b)**

### 5. Solution

Therefore, the solution to the system of equations is:

**x = a + b**

**y = -2ab / (a+b)**

**Note:** This solution is valid only if b ≠ 0 and a ≠ -b.