Question

How do you solve `\frac { 6y } { y ^ { 2} - 25} + \frac { 9} { y - 5} = \frac { 5} { y + 5}`?

Answer 1

`y=-7`

Explanation:

First, we should try to get the common denominator for all of the fractions.

We should know that `(y-5)(y+5)=y^2-25`. Therefore, we multiply the second fraction `(9/(y-5))` by `(y+5)` to get `((9y+45)/(y^2-25))`. The third fraction should be multiplied by `y-5` to make the denominator all the same, so we get: `5y- 25`.

Since all of the denominators are the same, we can ignore them for now to get....

`6y+9y+45=5y-25`.

We subtract `5y` on both sides and subtract `45` (in order to find what y is) and combine like terms...

`10y=-70`.

Divide 10 on both sides to get...

`y=-7`

You can plug it in if you want to, and you should get...

`-30/12=-5/2`.

If you need me to go through some details, just ask me in the comments.

-Sakuya

Answer 2

`y=-7`

Explanation:

Notice that the first denominator is `y^2-25`

This is the same as `y^2-5^2->(y-5)(y+5)` so this is our starting point.

`(6y)/((y-5)(y+5))+9/(y-5)=5/(y+5)`

`color(green)([(6y)/((y-5)(y+5))]+[9/(y-5)]=[5/(y+5)]`

Multiply by 1 and you do not change the value. However, 1 comes in many forms.

`color(green)([(6y)/((y-5)(y+5))]+color(white)("dd")[9/(y-5)color(red)(xx1)]color(white)("dd")=color(white)("dd")[5/(y+5)color(red)(xx1)]`

`color(green)([(6y)/((y-5)(y+5))]+[9/(y-5)color(red)(xx(y+5)/(y+5))]=[5/(y+5)color(red)(xx(y-5)/(y-5))]`

`(6y)/((y-5)(y+5))color(white)("dd")+color(white)("d") (9y+45)/((y-5)(y+5) )color(white)("d")=color(white)("d") (5y-25)/((y-5)(y+5)`

Now that the denominators are all the same we can just forget about them. Or, if you wish to be a purist you can say: multiply all of both sides by `(y-5)(y+5)`

`6y+9y+45=5y-25`

Collecting like terms we have:

`6y+9y-5y=-25-45`

`10y=-70`

divide both sides by 10

`y=-7`