Question

(a – b)x + (a + b)y = a² – 2ab – b², (a + b)(x + y) = a² + b² gives the following solution:

Options

• `x = a + b, y = (-2ab)/(a + b)`

• `x = (-2ab)/(a + b), y = a + b`

• `x = a + b, y = (2ab)/(a + b)`

• `x = (2ab)/(a + b), y = a + b`

Answer 1

`bb(x = a + b, y = (-2ab)/(a + b))`

Explanation:

Step 1: Simplify Equations

The given system of equations is:

1. (a – b)x + (a + b)y = a² – 2ab – b²
2. (a + b)(x + y) = a² + b²

Expand the second equation:

(a + b)x + (a + b)y = a² + b²

Step 2: Eliminate the y Variable

Subtract the first equation from the simplified second equation to eliminate the y term:

[(a + b)x + (a + b)y] – [(a – b)x + (a + b)y]

= [a² + b²] – [a² – 2ab – b²]

Simplify the expression:

(a + b – (a – b))x = a² + b² – a² + 2ab + b²

(a + b – a + b)x = 2ab + 2b²

(2b)x = 2b(a + b)

Step 3: Solve for x

Assuming b ≠ 0, divide both sides by 2b:

`x = (2b(a + b))/(2b)`

x = a + b

Step 4: Solve for y

Substitute the value of x into the simplified second equation:

(a + b)x + (a + b)y = a² + b²

(a + b)(a + b) + (a + b)y = a² + b²

(a + b)² + (a + b)y = a² + b²

a² + 2ab + b² + (a + b)y = a² + b²

Subtract a² + b² from both sides:

2ab + (a + b)y = 0

(a + b)y = –2ab

Assuming a + b ≠ 0, divide both sides by (a + b):

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