Question

Solve the following system of equations by the elimination method:

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

Answer 1

Given system:

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

(a + b) (x + y) = a² + b²    ...(2)

Step 1: Expand Equation (2)

Expanding equation (2):

(a + b)x + (a + b)y = a² + b²

Step 2: Write both equations for elimination

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

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

Step 3: Subtract equation (1) from equation (2)

Subtract (1) from (2) to eliminate (y):

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

= (a² + b²) – (a² – 2ab – b²)

Calculate coefficients:

(a + b) – (a – b) = a + b – a + b

(a + b) – (a – b) = 2b

(a + b) – (a + b) = 0

Calculate right side:

a² + b² – a² + 2ab + b² = 2b² + 2ab

So, 2bx = 2b² + 2ab.

Dividing both sides by (2b), assuming (b ≠ 0):

x = b + a

x = a + b

Step 4: Substitute (x = a + b) into equation (2) to find (y)

Recall equation (2):

(a + b)(x + y) = a² + b²

Put (x = a + b): 

(a + b)(a + b) + y = a² + b² 

(a + b)(a + b + y) = a² + b²

Divide both sides by (a + b) assuming (a + b ≠ 0): 

`a + b + y = (a^2 + b^2)/(a + b)`

Solve for (y):

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

Simplify the right-hand side:

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

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

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