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.
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