The one idea behind everything
Arithmetic deals with particular numbers: 3 + 4 is 7, and that is the end of the story. Algebra begins the moment we notice a *pattern* that holds no matter which numbers we use, and we want to write that pattern down once. To do it, we let a single letter — a variable — stand in for a number we have not chosen yet. Writing n + 1 means “take whatever number n is, and add one.” It is a sentence with a blank in it.
This is generalization: one symbolic statement that covers infinitely many arithmetic facts at once. Instead of checking that 2·3 equals 3·2, and 5·7 equals 7·5, and so on forever, we write a·b = b·a and capture every such fact in a single line.
Why bother with letters?
Three reasons. First, economy: one line replaces an endless list. Second, discovery: once a rule is written in symbols, you can rearrange the symbols and find new truths you never tested by hand. Third, honesty about the unknown: when you do not yet know a number but you do know a fact about it, a letter lets you write the fact down and *then* solve for the number.
Arithmetic (one fact at a time): 2 + 3 = 3 + 2 10 + 7 = 7 + 10 41 + 6 = 6 + 41 ... and on forever Algebra (all of them, one line): a + b = b + a Read it as: for ANY numbers a and b, adding in either order gives the same result.
The marks we use — letters, +, =, parentheses — make up algebraic notation. Each is a mathematical symbol with an agreed meaning. By convention letters near the end of the alphabet (x, y, z) usually mark quantities we are solving for, while letters near the start (a, b, c) usually mark fixed but unspecified amounts. Nothing forces this; it is just a shared habit that makes formulas easier to read.