Question 5698: Hi, I actually have 2 questions relating to adding and subtracting fractions using variables.
1)x-y/x+y + x+y/x-y
I thought it was 1 but now it seems that would be too easy.
2) 1/x - 1/x+4 = 1/3
*I know I'm supposed to find the LCD and multiply both sides by it, but I'm not sure what that is.
Answer by guapa(62)   (Show Source):
You can put this solution on YOUR website! 1) You have to find the LCD which is (x+y)(x-y). Now you have to multiply each fraction with the missing term:
(x-y)(x-y)/(x+y)(x-y) + (x+y)(x+y)/(x-y)(x+y)
With a common denominator in both fractions we can rewrite as follows:
(x-y)(x-y)+(x+y)(x+y)/(x+y)(x-y) Now let's simplify the numerator
(x^2-2xy+y^2)+(x^2+2xy+y^2)/(x+y)(x-y) Combine like terms in the numerator
(2x^2+2y^2)/(x+y)(x-y)
This is already your final answer. You can factor out the 2 in the numerator and evaluate the denominator but it won't change the value.
2(x^2+y^2)/(x^2-y^2) The sum of two squares is not factorable unless there is a common factor.
2)Like you already said you have to find the LCD which is 3x(x+4) in this case. Let's look how we found the LCD. Look at every denominator: x , (x+4) , 3
The LCD contains each different (prime) factor the most times it appears. Thus 3x(x+4).Now we have to multiply each fraction(numerator and denominator) by the missing term to produces equivalent fractions with the same denominator.
1/x-1/(x+4) = 1/3
{1*3(x+4)}/{x*3(x+4) - {1*(3x)}/{(x+4)*3x} = {1*x(x+4)}/{3*x(x+4)}
A big advantage when computing equations with fractions is that by using the LCD you can clear the equation of all fractions.
3(x+4) - 3x = x(x+4) Simplify both sides
3x+12-3x = x²+4x Combine like terms
x²+4x-12=0 To solve this equation we have to factor out the GCF. First we need to find 2 numbers whose sum is 4 and whose product is -12. 6 and -2 satisfy options.
x²+6x-2x-12=0 Now we can factor out
x(x+6)-2(x+6)=0
(x-2)(x+6)=0
Solve for x
x-2=0, x=2
x+6=0, x=-6
Check:
1/2 - 1/(2+4) = 1/3
1/2 - 1/6 = 1/3
3-1 = 2, 2=2
and
1/(-6) - 1/(-6+4) = 1/3
1/(-6) - 1/(-2) = 1/3
-1+3=2, 2=2
Hope it helps
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