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

##### 3 Answers

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:

Multiply the expression by 1 in the form of

Multiply the two fractions:

Substitute

Distribute y:

Distribute 12 and 18:

Combine like terms:

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

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

Please observe that

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

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