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

May 12, 2015

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

The idea is that fraction $\frac{a}{b}$ is equal to $\frac{m}{n}$ exactly when $a n$bm#.

So, to solve

$\frac{x + 10}{x - 6} = \frac{x}{x + 4}$

We cross multiply, to get:

$\left(x + 10\right) \left(x + 4\right) = x \left(x - 6\right)$

So, ${x}^{2} + 4 x + 10 x + 40 = {x}^{2} - 6 x$

${x}^{2} + 14 x + 40 = {x}^{2} - 6 x$

Sutracting ${x}^{2}$ from both sides gets us

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

$20 x = - 40$, 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

$\frac{\left(- 2\right) + 10}{\left(- 2\right) - 6}$ is equal to $\frac{- 2}{\left(- 2\right) + 4}$.