# Introduction

Elementary algebra is a system for talking about numbers without having to say, or even know, exactly what the numbers are. You do this by using some other symbol to stand for each number. This symbol could be anything that doesn't already have a meaning, usually a letter like x, A, or n. You can also use letters from other languages, like φ (Greek) or ℵ (Hebrew), entire words like `length` or `population`, or even nonsense symbols like ↵ or ◊. I'll mostly use ordinary English letters.

So if you want to say that it doesn't matter what order you add two real numbers, then rather than saying 3 + 4 = 4 + 3 (since they're both 7) and 2 + 9 = 9 + 2 (since they're both 11) and so on, you can just say that

a + b = b + a
for all real numbers a and b. Or if all you know about a real number is that you get 5 when you add 2 to it, then you can say that
x + 2 = 5
and see if you can figure out what x is; as it turns out, x can and must be 3. Or if you know that the length of a box is twice its width, then you can write this as
l = 2w,
where l stands for the length and w stands for the width. These examples don't really need algebra, because you can deal with them using ordinary words. But more complicated problems really need symbols, or you'll get lost.

Probably the most important technique to use in algebra is substitution. If two things are known (or assumed) to be equal, then you can replace one of them with the other in any expression. For example, you know that 3 + 7 = 10, so 3 + 7 + 5 = 10 + 5; you also know that 10 + 5 = 15, so 3 + 7 + 5 = 15. Similarly, if (in some problem) you know that x = 3, then x + 5 = 3 + 5; you also know that 3 + 5 = 8, so x + 5 = 8. Finally, if you assume that x = 7, then the inequlity x − 2 < 5 means the same thing as the inequality 7 − 2 < 5; since 7 − 2 = 5 and it is false that 5 < 5, it must also be false that x − 2 < 5. If you know that x − 2 < 5 really, then your assumption that x = 7 must also be false.

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