Question

What is the solution to the proportion `(x+10)/(x-6) = x/(x+4)`?

Answer 1

When solving a proportion problem (like this problem), one approach is to "cross multiply".

The idea is that fraction `a/b` is equal to `m/n` exactly when `an`bm#.

So, to solve

`(x+10)/(x-6) = x/(x+4)`

We cross multiply, to get:

`(x+10)(x+4) =x (x-6)`

So, `x^2+4x+10x+40 = x^2-6x`

`x^2+14x+40=x^2-6x`

Sutracting `x^2` from both sides gets us

`14x+40=-6x`. Adding `6x` and subtracting `40` on both sides yields:

`20x=-40`, so `x=-2`

We are not quite finished. Because we multiplied by expressions involving a variable, we need to.make sure we did not multiply by `0`.

When `x=-2`, neither `x-6` nor `x+4` is zero, so we should be ok.

As a final check make sure that

`((-2)+10)/((-2)-6)` is equal to `(-2)/((-2)+4)`.