Date: 25 Apr 1995 17:23:39 -0400 From: Sandip Mody Subject: Fractions and Fractional Equations Hello, I was having trouble with the following problems: 4/x-3 + 6/x+3 = 2/3 and solving for x F= w/x+1 The other problem was a^2 + b^2/a^2-b^2 / a-b/a+b - a+b/a-b Thank You for your help.
Date: 26 Apr 1995 11:15:09 -0400 From: Dr. Ethan Subject: Re: Fractions and Fractional Equations Hello, Good questions. I will assume that you know how to work with fractions when there are just numbers. If so, then this will be easy because here we do the same thing. What I will do is work the first and last problem; then I will leave the second problem for you to think about. Okay? Great. The first problem is: 4/x-3 + 6/x+3 = 2/3 Well the thing that we need to do is find a common denominator for the first two fractions. That will be (x-3)(x+3), so we need to multiply the first fraction by (x+3)/(x+3) and the second by (x-3)/(x-3). This will not change the value of the equation because both of these reduce to one. Now we have: 4(x+3)/(x+3)(x-3) + 6(x-3)/(x+3)(x-3) = 2/3 which becomes via addition: (4(x+3) + 6(x-3))/(x^2 - 9) = 2/3 Now, we multiply both sides by (x^2 - 9) to get: 4(x+3) + 6(x-3) = 2(x^2 - 9)/3 Now simplify the left hand side and we have: 10x - 6 = 2(x^2 - 9)/3 Now multiply both sides by 3 and we have: 30x - 18 = 2x^2 - 18 the 18's cancel and we are left with: 30x = 2x^2 Divide by two on both sides: 15x = x^2 So both 0 and 15 are solutions to this problem. Yea! Now to your last problem. Here it is. I am adding () where I think that you meant to put them. If I am wrong then write back and I will rework the problem. ((a^2 + b^2)/(a^2 - b^2))/((a-b)/(a+b) - (a+b)/(a-b)) Let's work with the bottom first. Since we have subtraction we need to find a common denominator. Again it will be (a+b)(a-b), so the first term on the bottom needs to be multiplied by (a-b)/(a-b) and the second by (a+b)/(a+b) so it looks like this: ((a^2 + b^2)/(a^2 - b^2))/((a-b)(a-b)/(a-b)(a+b) - (a+b)(a+b)/(a+b)(a-b)) this adds together to be: ((a^2 + b^2)/(a^2 - b^2))/(((a-b)^2 - (a+b)^2)/a^2 - b^2 Now square the subtracted terms and combine them and we have: ((a^2 + b^2)/(a^2 - b^2))/(-4ab/(a^2-b^2)) Now we can consider the whole fraction, and we can do the old invert and multiply trick for dividing fractions so the division problem above looks like this now: (a^2 + b^2) a^2 - b^2 ------------- * ----------- a^2 - b^2 -4ab Well the a^2-b^2 terms cancel and we are left with: -(a^2 + b^2)/4ab as our answer. I hope that I haven't made a careless mistake. I have tried to explain each step. If you need more explanation or you think that I have made a mistake, then you can write back. Also, if you need help on that other problem or have other questions please write to Dr. Math. Ethan, Doctor On Call
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