Equivalent Expressions

3n + 2n is 5n, because three of something plus two more of it makes five.

Only like terms combine

3n + 2 does not simplify. n and 1 are different things, exactly as fifths and thirds were.

Distributing

4(n + 3) means 4 lots of (n + 3), which is 4n + 12.

Testing two expressions

If two expressions are equivalent, any value put into both gives the same answer.

Two recipes, always the same result

2(n + 3) and 2n + 6 are equivalent: whatever number you substitute, both give the same answer. Equivalence means agreement for every value, not just the one you happened to test.

Testing is evidence, not proof

Substituting n = 5 into both and getting 16 both times is good evidence. It is not proof — two expressions can agree at one value and differ elsewhere. Proof comes from transforming one into the other using the properties.

The transformations that are allowed

The commutative, associative and distributive properties let you rewrite an expression without changing what it computes. Every legitimate step in algebra is one of these applied. Knowing their names makes it possible to say why a step was allowed.

Why you would rewrite an expression

One form may be easier to compute, another easier to understand, another easier to solve. 2(n + 3) shows the structure; 2n + 6 is easier to combine with other terms. Choosing the useful form is a skill in its own right.