Math 1 · Lesson 5 · part 1 of 5
The Words We Use
The vocabulary for this lesson, in one place.
Words we use
| Word | Plain English | What you do |
|---|---|---|
| Equation | A balance that says two sides are equal, like 5x + 2 = 12. | Keep both sides equal at every step. That is why every move in Step 3 happens on both sides. |
| Expression | A piece of math with no equals sign, like 5x + 2. | You cannot solve it, because there is nothing to balance. You can only tidy it up with Step 1 and Step 2. |
| Translate (Step 1) | Lesson 3's word for Step 1. Rewrite the line so it is only addition and multiplication, with the invisible 1s written in. | Do it first, every single time, before you distribute or combine anything. |
| Like terms | Terms with the exact same letter part. 7x and 2x match that way, and so do 1x and (−1)x. 6x and 6 do not, because 6 has no letter at all. | Hunt for them first. Only terms with the exact same letter part can be combined into one term. Terms that do not match just sit side by side. |
| Coefficient | The number on the variable. It means the variable is multiplied by that number, so 3x is 3 times x. | Write it in, even when it is 1. A lonely x is 1x, and −x is (−1)x. What you leave in your head is what you get wrong. |
| Constant | A plain number with no letter on it, like the 5 in 3x + 5. | A plain number combines only with another plain number. There is no letter to pull out, so you just add them. |
| Combining like terms | Putting terms with the same letter part into one term. You add the coefficients (the numbers on the variables) and keep the letter. | 5x + 2x = (5 + 2)x = 7x. Add the numbers, never add the letter. Write the invisible 1s in first, so x + 4x is 1x + 4x = 5x. |
| Commutative property of addition | The terms trade places. So 2 + 5x is the same as 5x + 2. | Use it in Step 2 to slide like terms (terms with the exact same letter part) next to each other. |
| Associative property of addition | The terms change partners, the grouping moves. Lesson 2 drew this as the parentheses moving. | Use it in Step 2 to group like terms (terms with the exact same letter part), and to group two plain numbers. |
| Associative property of multiplication | The factors change partners, the grouping moves. So 16 × 6x is the same as (16 × 6)x. | Use it on the last move of Step 3. It puts the multiplicative inverse (the reciprocal) and the coefficient (the number on the variable) together, and those two make 1. |
| Distributive property | Hand one out to everyone. So 3(x + 4) = 3x + 12. | Use it at the start of Step 2, whenever there are parentheses. It works over a subtraction too, which Lesson 2 taught: 3(x - 4) = 3x + (−12). |
| Distributive property run backwards | Factoring, pulling out what the terms share, from Lesson 2. Forwards it hands a number out, backwards it pulls the shared letter out. | Name it every time you combine like terms (terms with the exact same letter part): 5x + 2x = (5 + 2)x = 7x. |
| Additive inverse | The opposite. The opposite of 4x is −4x, and the opposite of 6 is −6. | Add it to both sides to make a term go away. The term and its opposite make 0. |
| Additive identity | The do-nothing number 0. Adding 0 changes nothing. | It turns up every time a term meets its additive inverse (the opposite). Then 0 + 13 = 13, and you just write 13. |
| Multiplicative inverse | The reciprocal, the flipped-over number. The reciprocal of 23 is 32. | Multiply both sides by it on the last move of Step 3. Lessons 1 to 4 called it the undo partner. |
| Multiplicative identity | The do-nothing number 1. Multiplying by 1 changes nothing. | It turns 1x into x on the last line. It is also why a lonely x is 1x. |
| The opposite rule | The opposite of x is (−1)x, so −x is really (−1)x. | Write the −1 in during Step 1. Do not credit the multiplicative identity (the do-nothing number 1) for this move, because it is its own rule. |
| The zero rule | 0 times anything is 0, so 0x is 0. | Use it when two x terms cancel: 2x + (−2x) = (2 + (−2))x = 0x = 0. |
| Addition property of equality | Add the same thing to both sides. Lesson 4 called this doing the same thing to both sides. | Use it twice in Step 3: once to move an x term, once to move a plain number. |
| Multiplication property of equality | Multiply both sides by the same thing. It is Lesson 4's balance idea with a formal name. | Use it on the last move of Step 3. Multiply both sides by the multiplicative inverse (the reciprocal) of the coefficient (the number on the variable). |