Question

How do you solve `\frac { 5} { y - 2} = \frac { y } { 3}`?

Answer 1

See a solution process below:

Explanation:

First, multiply each side of the equation by `color(red)(3)color(blue)((y - 2))` to eliminate the fractions while keeping the equation balanced:

`color(red)(3)color(blue)((y - 2)) xx 5/(y - 2) = color(red)(3)color(blue)((y - 2)) xx y/3`

`color(red)(3)cancel(color(blue)((y - 2))) xx 5/color(blue)(cancel(color(black)(y - 2))) = cancel(color(red)(3))color(blue)((y - 2)) xx y/color(red)(cancel(color(black)(3)))`

`color(red)(3) xx 5 = color(blue)((y - 2)) xx y`

`15 = y^2 - 2y`

Next, put the equation in standard form:

`15 - color(red)(15) = y^2 - 2y - color(red)(15)`

`0 = y^2 - 2y - 15`

`y^2 - 2y - 15 = 0`

Then factor the left side of the equation as:

`(y - 5)(y + 3) = 0`

Now, solve each term on the left for `0`:

Solution 1:

`y - 5 = 0`

`y - 5 + color(red)(5) = 0 + color(red)(5)`

`y - 0 = 5`

`y = 5`

Solution 2:

`y + 3 = 0`

`y + 3 - color(red)(3) = 0 - color(red)(3)`

`y + 0 = -3`

`y = -3`

The Solutions Are:

`y = {-3, 5}`