Question

How do you simplify the expression `(y-12/(y-4))/(y-18/(y-3))`?

Answer 1

I chose to multiply by a form of 1 that eliminates the complex fractions but there may be a better way, because I, later, discover a common factor that becomes 1.

Explanation:

Simplify: `(y-12/(y-4))/(y-18/(y-3))`

Multiply the expression by 1 in the form of `((y-4)(y-3))/((y-4)(y-3))`:

`(y-12/(y-4))/(y-18/(y-3))((y-4)(y-3))/((y-4)(y-3))`

Multiply the two fractions:

`((y(y-4)(y-3))- 12(y-3))/((y(y-4)(y-3))- 18(y-4))`

Substitute `(y^2-7y+12)` for `(y-4)(y-3))`:

`(y(y^2-7y+12)- 12(y-3))/(y(y^2-7y+12)- 18(y-4))`

Distribute y:

`(y^3-7y^2+12y- 12(y-3))/(y^3-7y^2+12y- 18(y-4))`

Distribute 12 and 18:

`(y^3-7y^2+12y- 12y+36)/(y^3-7y^2+12y- 18y+72)`

Combine like terms:

`(y^3-7y^2+36)/(y^3-7y^2- 6y+72)" [1]"`

I asked WolframAlpha to factor the numerator and obtained the following answer:

`(y + 2) (y - 3) (y - 6)`

Then I asked WolframAlpha to factor the denominator and obtained the following answer:

`(y + 3) (y - 4) (y - 6)`

Please observe that `(y-6)` is common to both numerator and denominator, therefore, expression [1] becomes:

`((y + 2) (y - 3))/((y + 3) (y - 4))`

Answer 2

`(y-12/(y-4))/(y-18/(y-3)) =((y+2)(y-3))/((y+3)(y-4))`

Explanation:

We want to simplify:

`(y-12/(y-4))/(y-18/(y-3))`

Firstly consider the numerator, which we can simplify by putting over a common denominator, thus:

`y-12/(y-4) = (y(y-4)-12)/(y-4)`
`" " = (y^2-4y-12)/(y-4)`
`" " = ((y+2)(y-6))/(y-4)`

Secondly consider the denominator, which we can also simplify by putting over a common denominator, thus:

`y-18/(y-3) = (y(y-3)-18)/(y-3)`
`" " = (y^2-3y-18)/(y-3)`
`" " = ((y+3)(y-6))/(y-3)`

So now we can rewrite the expression as:

`(y-12/(y-4))/(y-18/(y-3)) = {((y+2)(y-6))/(y-4)}/{((y+3)(y-6))/(y-3)}`

`" " = {((y+2)(y-6))/(y-4)} * {(y-3)/((y+3)(y-6))}`

Then cancelling the common factor of `(y-6)` we get:

`(y-12/(y-4))/(y-18/(y-3)) = {((y+2))/(y-4)} * {(y-3)/((y+3))}`

`" " = ((y+2)(y-3))/((y+3)(y-4))`

Answer 3

`color(magenta)(((y+2)(y-3))/((y+3)(y-4))`

Explanation:

`(y-12/(y-4))/(y-18/(y-3))`

`:.=((y(y-4)-12)/(y-4))/((y(y-3)-18)/(y-3))`

`:.=(y^2-4y-12)/(y-4) xx (y-3)/(y^2-3y-18)`

`:.=((y+2)(cancel(y-6)))/((y-4)) xx ((y-3))/((y+3)(cancel(y-6))`

`:.=color(magenta)(((y+2)(y-3))/((y+3)(y-4))`