## Solving the System of Equations: (a+c)x-(a-c)y=2ab and (a+b)x-(a-b)y=2ab

This article will guide you through the process of solving the system of equations:

**(a+c)x-(a-c)y=2ab**

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

We will utilize the **elimination method** to solve for the variables 'x' and 'y'.

### Step 1: Identify the Coefficients

Notice that the coefficients of 'y' in both equations are the same, but with opposite signs. This sets the stage for easy elimination.

### Step 2: Eliminate 'y'

Subtract the second equation from the first equation:

**(a+c)x-(a-c)y - [(a+b)x-(a-b)y] = 2ab - 2ab**

Simplifying the equation:

**(a+c)x - (a-c)y - (a+b)x + (a-b)y = 0**

**(a+c)x - (a+b)x - (a-c)y + (a-b)y = 0**

**(c-b)x = 0**

### Step 3: Solve for 'x'

Since the coefficient of 'y' has been eliminated, we can solve for 'x':

**x = 0 / (c-b)**

**x = 0**

Therefore, the value of 'x' is **0**.

### Step 4: Substitute 'x' to Find 'y'

Substitute the value of 'x' (0) into either of the original equations. Let's use the first equation:

**(a+c)(0) - (a-c)y = 2ab**

Simplifying the equation:

**-(a-c)y = 2ab**

**y = -2ab / (a-c)**

Therefore, the value of 'y' is **-2ab / (a-c)**.

### Conclusion

The solution to the system of equations is:

**x = 0**

**y = -2ab / (a-c)**

It's important to note that this solution is valid as long as **(a-c) ≠ 0**. If (a-c) = 0, then the system of equations would have no unique solution, as the two equations would represent the same line.