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

3 Answers
Jul 2, 2017

Answer:

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))#

Jul 2, 2017

Answer:

# (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)) #

Jul 2, 2017

Answer:

#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))#