Perform the indicated operations and reduce all results to lowest terms. Assume the variables are restricted to values that prevent division by 0.
(x^2-y^2)/(x^2-2xy+y^2) divided by (3x+3y)/(7x-21)
Could someone please show me how to do this so I can do the rest of my homework? I don't understand this at all. Thank You
(x^2-y^2) = (x+y)(x-y)
x^2-2xy+y^2 = (x-y)(x-y)
so when you divide the first by the second you get
(x+y)/x-y)
Now work on the bottom
(3x+3y) = 3(x+y)
(7x-21)= 7(x-3)
Now turn that upside down and multiply by the numerator
7(x-3) /3(x+y) * (x+y)/(x-y)
(7)(x-3)/3(x-y)
-1(y+6)+3(y+1)+7y
To divide two fractions, we can use the following rule: When dividing fractions, we actually multiply the first fraction by the reciprocal (or the inverse) of the second fraction.
Let's simplify the expression step by step:
Step 1: Simplify both the numerator and denominator of the first fraction.
Numerator: (x^2 - y^2)
This can be factored as (x - y)(x + y)
Denominator: (x^2 - 2xy + y^2)
This can be factored as (x - y)(x - y) or (x - y)^2
So, the first fraction becomes: (x - y)(x + y) / (x - y)^2
Step 2: Simplify both the numerator and denominator of the second fraction.
Numerator: (3x + 3y)
This can be factored as 3(x + y)
Denominator: (7x - 21)
This can be factored as 7(x - 3)
So, the second fraction becomes: 3(x + y) / 7(x - 3)
Step 3: Now, we'll multiply the first fraction by the reciprocal of the second fraction.
Reciprocal of 7(x - 3) is 1 / (7(x - 3))
Multiplying the fractions: (x - y)(x + y) / (x - y)^2 * (1 / (7(x - 3)))
Step 4: Simplify the result.
The (x - y) in the numerator and denominator cancels out, leaving:
(x + y) / (x - 3) * (1 / 7)
We can multiply the numerators together and the denominators together:
(x + y) / (7(x - 3))
And that's the simplified expression.