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I am attempting to learn integration by partial fractions but I really have no idea how to do any part of it so I need to learn to solve some complex equation like this first.

I really have no idea where to start, there isn't a common factor anywhere.

$17.5A + 10.5B + 7.5C = 43$

This looks a little bit like progress.

$2.3333333A + 1.4B + C = 5.733333$

This looks a little bit better maybe.

From here I think it involved a complex series of solving for 3 variables and I know those take several pages.

There has to be an easier way to do these problems otherwise the homework will take me several weeks.

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$2.33333$ is not right. No matter how many 3's you put after the 2., it is still not right. And that particular problem is not what matters in that post. And there are extra conditions that you are not specifying (that $A$, $B$, and $C$ be integers, because in that particular part of the post, we are talking about fractions with integer numerators and integer denominators). There are no series involving anything. – Arturo Magidin Jun 4 '12 at 20:19
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Note to all: this is a reference to this post; the equation in question appears in the introduction, which is not really about partial fraction decomposition, but rather the idea behind trying to do a partial fraction decomposition. Solving this equation in isolation is not something that would occur in a partial fraction decomposition, and is irrelevant to it. – Arturo Magidin Jun 4 '12 at 20:20
Why does it have to include integers? – Jordan Jun 4 '12 at 20:22
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Flagged this for closure as it is too localized unless @Jordan explains what exactly he wants. – user17762 Jun 4 '12 at 20:23
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@Jordan: In general, you need to get a clue first. One good method of getting clues is to listen. I've told you at least three times already that this particular equation and that part of the narrative is irrelevant for solving partial fraction decomposition problems, so the methods I used to solve it are irrelevant to you. Yet you don't listen. So, I'll stop talking. – Arturo Magidin Jun 4 '12 at 20:25
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closed as too localized by user17762, Arturo Magidin, MJD, Américo Tavares, t.b. Jun 5 '12 at 7:18

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2 Answers

up vote 2 down vote accepted

One can only solve the given equation for one variable, in terms of the others. When dealing with partial fraction decompositions, one typically has a system of equations to work with. If we had more equations with $A,B,C$ as variables, we might be able to solve for each of the three unknown constants.

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up vote 1 down vote

If A, B, C have to be [positive?] integers - consider that 86-21B must be a multiple of 5, which restricts options. Also 86-35A must be a multiple of 3, and 86-15C must be a multiple of 7. There are some obvious things to try.

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If they're positive integers you can probably solve it just by observing that $0≤A≤2$ and then handling the three cases one at a time. For instance, of $A=2$ then we are left with $21B+15C=16$, and this evidently has no solutions. If $A=1$, we get $21B+15C=51$, so $0≤B≤1$, etc. – MJD Jun 5 '12 at 0:02

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