Question

How do you solve `a/(ax-1)+b/(bx-1) = a+b` ?

Answer 1

`x = 0" "` or `" "x = (a^2+4ab+b^2)/(a^2b+ab^2)`

Explanation:

Given:

`a/(ax-1)+b/(bx-1) = a+b`

Subtract `a/(ax-1)+b/(bx-1)` from both sides to get:

`0 = a+b-a/(ax-1)-b/(bx-1)`

Multiply through by `(ax-1)(bx-1)` to get:

`0 = (a+b)(ax-1)(bx-1)-a(bx-1)-b(ax-1)`

`color(white)(0) = (a+b)(abx^2-(a+b)x+1)-a(bx-1)-b(ax-1)`

`color(white)(0) = (a+b)(abx^2-(a+b)x)+color(red)(cancel(color(black)((a+b))))-2abx-color(red)(cancel(color(black)((a+b))))`

`color(white)(0) = (a+b)(abx^2-(a+b)x)-2abx`

`color(white)(0) = x((a+b)(abx-(a+b))-2ab)`

`color(white)(0) = x((a+b)abx-((a+b)^2+2ab))`

`color(white)(0) = x((a^2b+ab^2)x-(a^2+4ab+b^2))`

`color(white)(0) = (a+b)abx(x-(a^2+4ab+b^2)/(a^2b+ab^2))`

Hence:

`x = 0`

or:

`x = (a^2+4ab+b^2)/(a^2b+ab^2)`