Question

Solve the following system of equations by the method of cross-multiplication:

`b/a x + a/b y = a^2 + b^2`

x + y = 2ab

Answer 1

Given: `b/a x + a/b y = a^2 + b^2, x + y = 2ab`

Step-wise calculation:

1. Put in standard form with coefficients:

`a_1 = b/a`

`b_1 = a/b`

`c_1 = a^2 + b^2`

a₂ = 1 

b₂ = 1

c₂ = 2ab

2. Compute the determinant

D = a₁b₂ – b₁a₂

`D = (b/a) xx 1 - (a/b) xx 1` 

= `b/a - a/b`

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

3. Compute `x = (c_1b_2 - b_1c_2)/D`: 

`c_1b_2 - b_1c_2 = (a^2 + b^2) xx 1 - (a/b) xx (2ab)` 

= a² + b² – 2a² 

= b² – a² 

Hence `x = (b^2 - a^2)/((b^2 - a^2)/(ab)) = ab`.

4. Compute `y = (a_1c_2 - c_1a_2)/D`: 

`a_1c_2 - c_1a_2 = (b/a) xx (2ab) - (a^2 + b^2) xx 1` 

= 2b² – (a² + b²) 

= b² – a² 

Hence `y = (b^2 - a^2)/((b^2 - a^2)/(ab)) = ab`.