(a-b)x+(a+b)y=a2-2ab-b2 (a+b)(x+y)=a2+b2 By Elimination Method

3 min read Jun 16, 2024
(a-b)x+(a+b)y=a2-2ab-b2 (a+b)(x+y)=a2+b2 By Elimination Method

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

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