Question

Solve the following system of equations by the elimination method:

x + y = a – b, ax – by = a² + b²

Answer 1

Given system of equations:

x + y = a – b   ...(1)

ax – by = a² + b²   ...(2)

Step 1: Multiply equation (1) by (b) to align coefficients for elimination:

b(x + y) = b(a – b)

⇒ bx + by = ab – b²   ...(3)

Step 2: Write down the two transformed equations:

bx + by = ab – b²   ...(3) 

ax – by = a² + b²   ...(2)

Step 3: Add equations (2) and (3) to eliminate (y):

(bx + by) + (ax – by) = (ab – b²) + (a² + b²) 

(bx + ax) + (by – by) = ab – b² + a² + b² 

(b + a)x = a² + ab

Step 4: Solve for (x):

`x = (a^2 + ab)/(a + b)` factor numerator.

x = a

Step 5: Substitute (x = a) into equation (1) to find (y):

a + y = a – b 

⇒ y = –b