Question

Solve the following two linear equation by substitution and elimination method: `ax+by=(a-b) , bx-ay=(a-b)` ?

Answer 1

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

Explanation:

`a*(ax+by)+b*(bx-ay)=a*(a-b)+b*(a-b)`

`a^2*x+aby+b^2*x-aby=a^2-ab+ab-b^2`

`(a^2+b^2)*x=a^2-b^2`

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

So,

`a*(a^2-b^2)/(a^2+b^2)+by=a-b`

`a*(a^2-b^2)+by*(a^2+b^2)=(a-b)*(a^2+b^2)`

`a^3-ab^2+(a^2+b^2)*by=a^3+ab^2-a^2*b-b^3`

`(a^2+b^2)*by=2ab^2-a^2*b-b^3`

`y=(2ab^2-a^2*b-b^3)/[b*(a^2+b^2)]`

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