For pre-algebra and beginning algebra students
Equality is one of the most important ideas in mathematics, and many students only half understand it. Once it gets the respect it deserves, everything else in algebra gets easier.
There's more to equality than writing x = 12. We need to know how to change an expression without changing its value, and how to change an equation without breaking its equality. Expressions have a value that sometimes has to be preserved. Equations have an equality that always must be.
Expressions and equations
An expression has no equals sign. Examples: 4, 3x, 3x + 2, 2x + 3x, 4x/5. Two kinds show up: constants like 4, whose value never changes, and those that depend on a variable like x. We count terms (separated by + or −): 3x has one, 3x + 2 has two. The word polynomial covers most of them, the way "car" covers sedans, trucks and SUVs.
An equation always has an equals sign: 2 + 3 = 5, or 2x + 6 = 10. Each side of an equation is an expression. That is what equations are made of.
Take 2x + 4x. Its value depends on x. If x = 6, it's 36. If x = 1, it's 6. We can't get rid of the x, but we can rewrite the expression in simpler form. Both 2x + 4x and 6x give the same value for any x, which follows from the distributive law: 2x + 4x = (2 + 4)x.
Solving one, step by step
Let's put those two expressions together into an equation. The goal is to get x alone on one side.
2x + 4x = 12
6x = 12 (simplify the left side)
x = 2 (divide both sides by 6)
In the first step, we worked on only one side. We rewrote 2x + 4x as 6x without changing its value, so the equality was preserved. In the second, we worked on both sides at once. You may add, subtract, multiply or divide both sides by the same amount, and the equality stays true. (Adding 5 to both sides would also be legal, but it wouldn't get us closer to the goal.)
Notice that both sides changed: 6x became x, and 12 became 2. That's fine. When you do the same thing to both sides, the equality in the original equation survives. And the final line tells us the answer: if x is 2, the original equation is true.
In short
When you simplify an expression, you preserve its value. When you do the same operation to both sides of an equation, you preserve its equality. That's the concept of equality at work, and it's the idea behind almost every step in algebra.