Question

How do you solve the system `[(x)/(x+5)]- [(5)/(5-x)] = (x+31)/(x^2-25)`?

Answer 1

Multiply numerators and denominators to get a common denominator `x^2-25` then equate the numerators to get a quadratic, hence `x = 3` or `x=-2`

Explanation:

`x/(x+5)-5/(5-x)`

`=x/(x+5)+5/(x-5)`

`=(x(x-5))/((x-5)(x+5)) + (5(x+5))/((x-5)(x+5))`

`=(x^2-color(red)(cancel(color(black)(5x)))+color(red)(cancel(color(black)(5x)))+25)/(x^2-25)`

`=(x^2+25)/(x^2-25)`

So we want to solve `x^2+25 = x+31` with exclusion `x != +-5`

Subtract `x+31` from both sides to get:

`0 = x^2-x-6 = (x-3)(x+2)`

So `x=3` or `x=-2`