I tried 4 times to solve this problem so that my solution matched the book solution. No luck. Here is the problem:
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Here is the book solution:
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Here is my solution and the tortured path I followed to reach it:
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I can't for the life of me figure out what I'm doing wrong here.
Several thoughts.
First, in theory, it makes no difference whether you work with
\(\displaystyle \dfrac{a}{b} - \dfrac{b}{c} + \dfrac{d}{e} \text { or } \dfrac{a}{b} + \dfrac{(-\ 1)b}{c} + \dfrac{d}{e}.\)
They are mathematically the same. However, I find that I make fewer sign errors if I avoid subtracting fractions: I always add fractions. To do so, I turn minus signs for fractions into plus signs by multiplying their numerators by - 1 before combining fractions.
So the very first thing I would do is
\(\displaystyle \dfrac{4}{y^2 - 3y + 2} - \dfrac{y}{y - 2} + \dfrac{2y}{y - 1} = \dfrac{4}{y^2 - 3y + 2} + \dfrac{-\ y}{y - 2} + \dfrac{2y}{y - 1}.\)
I doubt you would have made the error you did had you been adding fractions with the proper signs on the numerators. (I reirerate that this has nothing to do with math, but it seems to accord with human psychology.)
Second, check your work. A good way to check simplifications is to choose a small integer > 2, and see whether it works out. This is not 100% effective, but it will identify a very high percentage of errors. (If it is really important to be correct and you have the time, trying 2 different integers is almost certain to catch any errors.)
\(\displaystyle \dfrac{4}{3^2 - (3 * 3) + 2} - \dfrac{3}{3 - 2} + \dfrac{2 * 3}{3 - 1} =\)
\(\displaystyle \dfrac{4}{9 - 9 + 2} + \dfrac{-\ 3}{1} + \dfrac{2 * 3}{2} = \dfrac{4}{2} - 3 + 3 = 2.\)
\(\displaystyle \dfrac{-\ 3(3^2) + 5(3) + 4}{3^2 - (3 * 3) + 2} = \dfrac{-\ 3(9) + 15 + 4}{2} = \dfrac{-\ 8}{2} = -\ 4.\)
\(\displaystyle \text {AND } 2 \ne -\ 4.\)
So you do not need to look in the back of your book to know your answer is incorrect. Moreover, you know that the evaluation of each step should give the answer of 2 when 3 is substituted for y. This will let you find the exact step where you went wrong.