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Question 950256: rewrite the rational expression as an equivalent rational expression with the given denominator.
7a-3
------ = -------
8a+40 8y(a+5)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if you let that blank numerator = x, then you get:
(7a-3) / (8a+40) = x / (8y(a+5))
solve for x to get:
x = ((7a-3)*(8y(a+5)) / (8a + 40)
factor the denominator to get:
x = ((7a-3)*(8y(a+5)) / (8 * (a+5))
the 8 in the numerator and denominator cancel out.
the (a+5) in the numerator and denominator cancel out.
you are left with:
x = (7a-3) * y
your original equation becomes:
(7a-3) / (8a+40) = (7a-3)*y / (8y(a+5)
these two expressions should be equivalent.
to confirm, let a equal any number and let y equal any number and see if the equation is true.
for example:
when a = 5 and y = 8, you get:
you get .4 = .4 which means they are equivalent.
your solution is:
(7a-3) / (8a+40) = (7a-3)*y / (8y(a+5))
it doesn't matter that the y on the right side would cancel out, because you weren't asked to simplify your answer. you were only asked to make the left side of the equation equivalent to the right side with the denominator on the right side as shown.
that you did.
now that you're done with all the hard work, you can probably spot some relationship that maybe you didn't see before.
start with your original equation.
(7a-3) / (8a+40) = x / (8y(a+5)
8y(a+5) can also be shown as y * 8 * (a+5) which can be shown as y * (8a+40)
your original equation becomes:
(7a-3) / (8a+40) = x / (y(8a+40)
the only difference in the denominators is that the denominator on the right is multiplying the (8a+40) by y.
to keep the fractions the same, multiply the equation on the left by y/y to get:
(7a-3) / (8a+40) = y(7a-3) / (y(8a+40))
convert the 8a+40 back to 8(a+5) and you get:
(7a-3) / (8a+40) = y(7a-3) / (8y(a+5))
this is exactly the same as your original solution of:
(7a-3) / (8a+40) = (7a-3)*y / (8y(a+5))
note that y(7a-3) is exactly the same at (7a-3)*y.
the solution was definitely arrived at quicker the second time around, once you saw the relationship.
that's not always apparent at first glance though, so the exercise of solving for x was worthwhile.
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