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I know it is an elementary algebra question. But, is there a good reason for it being valid. Let's say we have $x+5=7$. We obviously know $x=2$, but if I can add any number on both sides and it still is valid. Ex: $x+5+3=7+3$. Simplifying to $x+8=10$. This is also valid and $x$ still equals $2$

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  • $\begingroup$ If two things are equal, then treating them equally should yield equal results, no? $\endgroup$ – Blue Nov 30 '15 at 23:09
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    $\begingroup$ For example, if "the President of the U.S. = Obama", then "wife of the President of the U.S. = wife of Obama". $\endgroup$ – user856 Nov 30 '15 at 23:10
  • $\begingroup$ @Rahul That makes so much sense! I don't know why, but when I was thinking about it in terms of variables and numbers it wasn't clicking. But when you said that statement it clicked. $\endgroup$ – James Smith Nov 30 '15 at 23:12
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Fantastic question! You have stumbled upon what is called an axiom, specifically an axiom about equality. An axiom is essentially something that we all agree to accept so that we can get on doing mathematics (quite frankly, I can't imagine a system in which adding equals to equals would yield two things that are unequal!).

This particular axiom is extremely old, and dates back to Euclid, from around 300BC. In his great book of geometry, Elements, he states his axioms and calls them "Common Notions," things he believe to be true that are taken completely at face value. If you too believe in his "common notions," well, then you'll believe everything he uses them to demonstrate!

The second common notion is

If equals are added to equals, then the wholes are equal.

It's possible that more logically-minded people have come along and found a way to prove such a thing (I don't honestly know), but some of the greatest minds in history have merely taken it on faith.

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Just common sense really. Suppose that Xerxes has a bag containing $x$ apples and Yvonne has a bag containing $y$ apples. You don't know how many apples they have, but you do know that they are both the same: $$x=y\ .$$ If you now give each of them another two apples, then you still don't know how many each has, but you do know that they are still the same: $$x+2=y+2\ .$$

Comment: a more formal answer would be that this is one of the assumed properties of equality in predicate logic. But basically, the only reason it's assumed is because people think it's obvious, so that kinda comes back to the same answer...

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  • $\begingroup$ Thanks David, I love the simplicity. I also like Rahul in saying "if the President of the U.S. = Obama", then "wife of the President of the U.S. = wife of Obama!"...I don't know why, but it wasn't clicking in my head before. It seems pretty obvious now that you explain it. $\endgroup$ – James Smith Nov 30 '15 at 23:15
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    $\begingroup$ Considering it "common sense" goes all the way back to Euclid, who listed this as a "Common Notion". $\endgroup$ – Blue Nov 30 '15 at 23:15
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    $\begingroup$ @Majmun WHAT? $2+4=6$ and $2+1=3$.... $6$ doesn't equal $3$. Unless you are talking about if math were hypothetically defined addition differently. In that case I misunderstood you. Although I don't really think it would be useful if defined that way.**UPDATE**:Never mind he deleted his comment. $\endgroup$ – James Smith Nov 30 '15 at 23:22

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