# Simplify $-8x - 12 + 3x - 5x + 9$

So I'm playing around with some maths, looking through my old algebra and calculus books and came upon this problem and so, I went through and simplified it or at least I believe I've simplified it correctly.

Simplify $-8x - 12 + 3x - 5x + 9$

[(-8x - 12) + 3x] - 5x + 9                   order of operations
[-8x - (12 + 3x)] - 5x + 9                   associative property
at [... - (12 + 3x)], i think this is identity property
[-8x - (3x + 12)] - 5x + 9                   commutative property
[-8x - 3x - 12] - 5x + 9
-11x - 12 - 5x + 9                           combine like terms
[(-11x - 12) - 5x] + 9                       order of operations
[-11x - (12 - 5x)] + 9                       associative property
[-11x - (-5x + 12)] + 9                      commutative property
-11x + 5x - 12 + 9
-6x - 12 + 9                                 combine like terms


Is this the simplified answer -6x - 3 ?

Thanks,

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It should be $-10x-3$. The only thing you need to do is, $(-8+3-5)x+(9-12)$. – Gigili May 27 '12 at 18:09
ah, yes, even simpler. – developerdoug May 27 '12 at 18:18

Your error is in the second line:

$$(-8x-12x) +3x \neq -8x - (12x + 3x).$$ Addition associates, but subtraction does not: $(7-4)-3 = 0$, but $7-(4-3) = 7-1 = 6$.

The correct way to associate those terms is to remember that subtracting $12x$ is the same thing as adding $-12x$. So $$(-8x - 12x) + 3x = \Bigl((-8x)+ (-12x)\Bigr) + 3x = -8x + \Bigl((-12x) + 3x)\Bigr) = -8x + (-12x + 3x)$$ or by being careful with the $-$ sign: $$(-8x - 12x) + 3x = -8x - (12x - 3x)$$ (distribute the $-$ sign to see why they are equal).

Addition commutes and associates. All you need to do is consider the terms with $x$ together, and the constant terms together.

The terms with $x$ are $-8x$, $3x$, and $-5x$. When you add them together you get $$-8x+3x-5x = (-8+3-5)x = -10x.$$

The constant terms are $-12$ and $9$. When you add them together you get $-3$. So the final answer is $-10x-3$.

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awesome, thanks. – developerdoug May 27 '12 at 18:18