# simplify by multiplying by -1 ( Solving equations in terms of a variable)

I am trying to learn Algebra. I'm using Khan Academy, and I just "solved" this problem.

I understand all the steps, except #4.

Could someone please tell me why I would multiply by -1 to simplify the solution?

Mahalo in advance.

Solve for u.

uv−5uw−4u+9=8v−9

1. Combine constant terms on the right.

uv−5uw−4u+9=8v−9

uv−5uw−4u=8v−18

2. Factor out the u.

u⋅(v−5w−4)=8v−18

3. Isolate the u.

u=8v−18 / v−5w−4

4. We can simplify this by multiplying the top and bottom by −1.

u= 8v+18 / −v+5w+4

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In addition to Robert's answer, your step 4 yields an incorrect result. It should be $$u=\frac{-8v+18}{-v+5w+4},$$ instead. – Cameron Buie Jan 2 '13 at 7:42
Cameron, your formatting of Math looks great. Could you tell me how to do that, or point me in the right direction? – I-Ii Jan 2 '13 at 8:02
@Jason: take a look here. – Fabian Jan 2 '13 at 8:29
@Fabian, thank you. – I-Ii Jan 2 '13 at 8:32

## 2 Answers

It's not nearly as important as the other simplifications you've done. The general idea is just that dealing with positive terms is nicer than dealing with negative ones, so multiplying the top and bottom by -1 changes a lot of negative terms and a few positive terms into a lot of positive terms and a few negative terms, which would normally be seen as simpler. Some people might have left it out entirely in this case, since it doesn't really make a big difference here.

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That really clarifies it for me. It was all mystical, until I received your response. In addition, I foresee this multiplying by -1 trick to be useful in the future - as it is indeed easier to work with positives. – I-Ii Jan 2 '13 at 7:51

The last step, as reported, is incorrect. If we multiply top and bottom by $-1$, we should get $18-8v$ on top. Or the less attractive and potentially confusing $-8v+18$.

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Thank you for pointing that out. – I-Ii Jan 2 '13 at 7:50