Longer Equations: Combine, Then Solve

The big idea

There is nothing in a long equation that is not in a short one. Every line you write has a name, and you can say it out loud. Step 1 is translating, which is what Lesson 3 taught. Subtraction becomes adding the additive inverse (the opposite). A written ÷ sign becomes multiplying by the multiplicative inverse (the reciprocal). Every invisible 1 gets written in. Step 2 hands the outside number in, with the distributive property (hand one out to everyone). Then the commutative property of addition (the terms trade places) moves the like terms (the terms with the same letter part) together. The associative property of addition (the terms change partners, the grouping moves) groups them. Then the distributive property run backwards (factoring, pulling out what the terms share, from Lesson 2) combines them. Step 3 uses the addition property of equality (add the same thing to both sides). It ends with the multiplication property of equality (multiply both sides by the same thing). Those two are the grown-up names for what Lesson 4 called doing the same thing to both sides. Each move makes a do-nothing number appear: the additive identity (the do-nothing number 0), or the multiplicative identity (the do-nothing number 1). The opposite and the reciprocal are your undo partners from Lesson 1. A longer equation is just more of those lines, in the same order. If you can name the property on each line, you are not following steps you memorized. You are doing algebra, and you can prove every move you made.