Question

How do you simplify `\frac { x + 5y } { x + 3y } - \frac { 2x + y } { 3y + x } + \frac { 4x + 5y } { x + 3y }`?

Answer 1

`3`

Explanation:

Since all the denominators are equal, we can just combine the numerators together from
`(x+5y)/(x+3y)-(2x+y)/(3y+x)+(4x+5y)/(x+3y)` to

`(x+5y-(2x+y)+4x+5y)/(x+3y)`.

Let's work on the numerator first. First, we remove the brackets

`x+5y -2x-y +4x+5y`

Then group like terms.

`x-2x+4x+5y -y +5y`

`x+4x-2x =3x" "` and `" "5y-y+5y=9y`

The expression will now be `3x+9y`.

Put them all together as a fraction and you get `(3x+9y)/(x+3y)`.

Next, we will factor out the common term, which is `3`, to get

`(3(x+3y))/(x+3y)`.

We will cancel out `x+3y`.

`(3(cancel(x+3y)))/(cancel(x+3y))=3`.

The answer is `3`.