Don't Multiply by Zero
Date: 01/23/2002 at 21:36:48 From: Richard Subject: A meaningless answer from a fractional equation Here's the problem: 3y/y-7 - 3/2 = 21/y-7 solve for y. Here's what I did. I multiplied both sides of the equation by the LCD (Lowest Common Denominator), which is 2(y-7). After doing that, I got 6y - 3y + 21 = 42 Then I solved for y: 3y + 21 = 42 3y = 21 y = 7 But wait a minute, if I plug 7 in for all the y's, I get a total answer of 21/0 - 3/2 = 21/0. What kind of answer is this? Is it a meaningless solution?
Date: 01/23/2002 at 22:53:06 From: Doctor Peterson Subject: Re: A meaningless answer from a fractional equation Hi, Richard. It's not really a paradox; just a warning to be careful when you multiply. Remember that it is only legal to multiply both sides of an equation by a NON-ZERO value. So when you finish solving the problem, you have to go back and check whether 2(y-7) was zero. Since it was, the solution you found is not valid, and there is no solution to the equation. When this happens, I like to try to find another way to work it out without doing the illegal step. One approach is to simplify the left side by adding the fractions: 3y/(y-7) - 3/2 = 21/(y-7) 6y - 3(y-7) 42 ----------- = ------ 2(y-7) 2(y-7) Now we can subtract the right side, using the common denominator: 6y - 3(y-7) - 42 ---------------- = 0 2(y-7) 3y - 21 ------- = 0 2(y-7) 3(y-7) ------ = 0 2(y-7) Now, if y is not 7, we can further simplify this by canceling the common factor: 3/2 = 0 This is, as I hoped, an equation with no solution, obtained without doing anything questionable. It took a little more work this way, but gives me more confidence in saying there is no solution. I hope that gives you a little more confidence, too! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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