Express your answer in simplest form?

Question

Add: x² / x -5 + 4x – 45 / x -5

and

Subtract : 7 / y - 7 – 1 / y + 7

Divide: x²-36y² / 5x^2+30y ÷ (x² + 6xy)

I'm so very confused on how to work these problems can anyone show me the steps? Any assistance would be greatly appreciated!

ashriel lim & ironduke8159 thank you both for your help! It is greatly appreciated! Ya'll are awesome!

Answer 1

Add: x² /( x -5) + (4x – 45) / (x -5)
= (x^2+4x-45)/(x-5)
= (x+9)(x-5)/(x-5)
= x+9

Subtract : 7 /( y - 7) – 1 / (y + 7)
= (7y+49 -y+7)/(y^2-49)
=(6y+42)/[(y-7)(y+7)]
=6(y+7)/[(y-7)(y+7)]
=6/(y-7)

Divide: x²-36y² / 5x^2+30y ÷ (x² + 6xy)
Can't do anything with above. Are you sure you wrote it correctly?

Answer 2

Addition:
x² / x -5 + 4x – 45 / x -5

since you have a common denominator, you can add the numerators as they are:
[x² + 4x – 45] / [x-5]

You can still simplify the numerator:
[(x+9)(x-5)]/ x-5

You can cancel x-5, so what you’ll have is:
x+9.


Subtraction
7 / y - 7 – 1 / y + 7

You have different denominators so you have to get a common denominator. That would be (y-7)(y+7). You have to multiply the first numerator by (y+7) and the second numerator by (y-7). Therefore, you’ll have:
[7(y+7) / (y-7)(y+7)] – 1 (y-7) / (y-7)(y+7).

Simplifying, you get:
[7y+49 / (y-7)(y+7)] – [y-7 / (y-7)(y+7)]

Since you have a commond enominator, you can apply the operation between the numerators.
[(7y+49)-(y-7)] / [(y-7)(y+7)]

You’ll have:
[6y+ 42] / [(y-7)(y+7)]

You can factor the numerator:
6(y+7) / (y-7)(y+7)

Simplifying, you’ll have to cancel y+7:
6/ y-7


Division:
x²-36y² / 5x^2+30y ÷ (x² + 6xy)

Factor out everything first:
[(x-6y)(x+6y) / 5(x+6y)] ÷ x(x + 6y)

Simplifying the terms inside [ ], you can cancel x+6y. Therefore:
[x-6y / 5] ÷ x(x + 6y)
Division is the same as multiplying with the reciprocal, thus:
[x-6y / 5] [1/ x(x + 6y)]

Combining the term since nothing can be cancelled:
[x-6y] / [5x(x+6y)]

Answer 3

To Sarah, I would just like to correct,

In one of the solutions given, he have stated this:

Since you have a commond enominator, you can apply the operation between the numerators.
[(7y+49)-(y-7)] / [(y-7)(y+7)]

You’ll have:
[6y+ 42] / [(y-7)(y+7)]

--- I think you won't get 6y + 42 from this equation [(7y+49)-(y-7)]. Remember that there is still a minus sign which will be distributed to the eqaution y - 7 in parenthesis. So I think we'll get this:

[(7y+49)-(y-7)] = 7y+49-y+7
= 6y + 56 not 6y + 42

So, we won't be able to cancel it!

So with that, we'll have
6y + 56 / (y - 7) (y +7) = 2 (3y + 28) / (y - 7) (y +7)

I hope that helped!