# Why does this formula work? [closed]

While shopping, I was thinking about random stuff and realized this works. I don't really understand how I discovered it, but here it is. Works with every number. It would be helpful if someone can simplify it for me.

$2x = 10 + (x - (10 - x))$

I do, however, vaguely understand that it is finding the average of some number that $+10 = 2x.$

## closed as unclear what you're asking by Shailesh, choco_addicted, Vladimir Reshetnikov, S.C.B., user91500Jan 27 '17 at 6:33

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Just do the algebra! – TheGeekGreek Jan 26 '17 at 21:47
• Tried, got messed up. – TigerGold Jan 26 '17 at 21:47
• @TheGeekGreek... Yes. But the only reason Tiger would ask the question is he does not know algebra. – GEdgar Jan 26 '17 at 21:48
• Actually, I do. – TigerGold Jan 26 '17 at 21:49
• @TigerGold Yes, this is basic, so I assumed you know it. If you got stucked, see my answer below :) – TheGeekGreek Jan 26 '17 at 21:50

## 3 Answers

Here's an intuitive explanation. Suppose you start at the number $10$ and move up by $x$ units, arriving at $10+x$; that number is $x$ units away from $10$, right?

On the other hand, suppose you start again at the number $10$ and move down by $x$ units, arriving at $10-x$; that number is also $x$ units away from $10$, but in the exact opposite direction.

$$(10-x)\underbrace{\overbrace{\longleftarrow}^{x\textrm{ units}} 10\overbrace{\longrightarrow}^{x\textrm{ units}}}_{2x\textrm{ units}} (10+x)$$ It's like saying that if you are standing in the exact center of town, and the western border is a mile from you, and the eastern border is a mile from you, then the two borders are two miles apart.

So these two final numbers must be $2x$ units away from each other. The distance between the two final numbers is just their difference, so we are saying that $$2x = (10+x)-(10-x)$$ That's the same thing as $10+(x-(10-x))$, as you can easily test with some specific numbers.

• That explains better! Thank you. – TigerGold Jan 26 '17 at 21:58
• You're welcome. Note that "10" is irrelevant here, because the same thing happens no matter where you start. So we can say in general that $$2x = y+(x-(y-x))$$ since $y$ plays the same role as 10 in the original. – MPW Jan 26 '17 at 22:04
• I know, I discovered that too, just used it as an easy number. – TigerGold Jan 27 '17 at 1:07

$$2x = 10 + (x - (10 - x))$$ First note that $-(10-x)$ is just a shorter way of saying $-1 \cdot (10 - x)$. So what we really have is $$2x = 10 + (x - 1 \cdot (10-x))$$ Now we can use the distributive law and associativity law, along with some simplification, to get \begin{align*} 2x &= 10 + (x \color{blue}{- 1} \cdot (\color{green}{10}\color{red}{-x}))\\ 2x &= 10 + (x \color{blue}{-1} \cdot \color{green}{10} \color{blue}{-1} \cdot (\color{red}{-x})) \qquad \text{use distributive law}\\ 2x &= 10 + (x - 10 + x) \qquad\qquad\qquad \text{simplify}\\ 2x &= 10 + (2x - 10) \qquad\qquad\qquad\quad \text{simplify some more}\\ 2x &= 10 + 2x - 10 \qquad\qquad\qquad\quad \text{drop parentheses (associativity law)}\\ 2x &= 2x \qquad\qquad\qquad\qquad\qquad\qquad \text{simplify} \end{align*}

• Nice answer, but I think the coloring in this case is a bit overused: three colors is over kill; at least I found it to be distracting, rather than illuminating. – Namaste Jan 26 '17 at 22:04
• @amWhy thanks. I waffled on the colors for a bit and figured I wouldn't be perfectly happy either way so I just chose one and ran with it. Eh.. – tilper Jan 27 '17 at 3:55

As summations are associativ, you can simply delete the outer bracket without changing the value of the right-hand side.

So we have $$2x=10 +x -(10-x)$$

The remaining one can be deleted if you interchange the signs of the summands inside of it (Do you know why?). Then your formula becomes $$2x=10+x-10+x.$$

It is easy to see that this equation holds.

• Ok. I see, I was dumb for not noticing that. – TigerGold Jan 26 '17 at 21:50
• Yu were interested, this is never a bad thing. – klirk Jan 26 '17 at 21:57