Question

How do you solve `\frac { 8} { x + 9} = \frac { 9} { x }`?

Answer 1

See a solution process below:

Explanation:

First, multiply each side of the equation by `color(red)(x)color(blue)((x + 9))` to eliminate the fractions while keeping the equation balanced:

`color(red)(x)color(blue)((x + 9)) xx 8/(x + 9) = color(red)(x)color(blue)((x + 9)) xx 9/x`

`color(red)(x)cancel(color(blue)((x + 9))) xx 8/color(blue)(cancel(color(black)(x + 9))) = cancel(color(red)(x))color(blue)((x + 9)) xx 9/color(red)(cancel(color(black)(x)))`

`8x = 9color(blue)((x + 9))`

`8x = (9 xx color(blue)(x)) + (9 xx color(blue)(9))`

`8x = 9x + 81`

Now, subtract `color(red)(8x)` and `color(blue)(81)` from each side of the equation to solve for `x` while keeping the equation balanced:

#8x - color(red)(8x) - color(blue)(81) = 9x - color(red)(8x) + 81 - color(blue)(81)#

`0 - color(blue)(81) = (9 - color(red)(8))x + 0`

`-81 = 1x`

`-81 = x`

`x = -81`