Math 1 · Lesson 4
Solving Equations
Solving is just undoing. But before you undo anything, do Step 1 every time: rewrite subtraction and division, and write in the invisible 1s. Then undo whatever is added to the letter and its coefficient, keeping both sides balanced.
Step 1, before you solve anything: rewrite
Every equation starts with the same move. Do both parts, every time, before you touch anything else.
- AGet rid of subtraction and divisionTurn every subtraction into adding the opposite (the additive inverse), and every division into multiplying by the reciprocal (the multiplicative inverse). Now the whole equation is only addition and multiplication.
WhySubtraction and division do not follow the properties of algebra (Lesson 2). They do not commute (5 − 2 is not 2 − 5), they do not associate, and you cannot distribute over them. Addition and multiplication do all of that. Once everything is + and ×, every rule you know is allowed.
- BWrite the invisible 1sThe coefficient is the number on the variable: it means the variable is multiplied by that number, so 3x is 3 times x. A lonely x is 1x (the coefficient is 1), and −x is (−1)x. Write them in.
WhyWe usually do not write the 1, and anything algebra leaves you to do in your head is one of the top causes of mistakes. 5x − x is 4x, not 5, and the only reason people get 5 is the 1 they never wrote down.
In the practice below, the Step 1 rewrite is shown in this color.
To solve: get x alone
To solve an equation means to get the letter by itself, so the last line reads x = the answer. An equation is a balance, so whatever you do to one side you must do to the other. After Step 1, there are only two things that can be on the letter: a number added to it, and its coefficient. Undo them in that order.
- 1RewriteStep 1. No subtraction, no division, every 1 written in.
- 2Add the oppositeof any number that is added to x, on both sides. That number and its opposite make 0, the do-nothing number, so it disappears.
- 3Multiply by the reciprocalof the coefficient on x, on both sides. The coefficient and its reciprocal make 1, the do-nothing number, so x is alone.
- 4CheckPut your answer back into the original equation. Both sides should match.
Why that order? Think of getting dressed: socks first, then shoes, so to undo it the shoes come off first. The number added on is the outside layer. The number you add or multiply by is shown in this color.
Example 1: 2x − 6 = 8
Example 2: x ÷ 5 + 1 = 3
A fraction is not a division. x3 is not "x divided by 3." It is 13x: a coefficient of 13 on x. Only a written division sign (÷) is a division to rewrite. If there are parentheses, such as 3(x + 2) = 21, distribute first (hand the 3 to everyone inside, Lesson 2) and then it is an ordinary equation.
Warm-up: the three moves
Four equations. For each one: rewrite first (Step 1), then add the opposite of the number added to x (if there is one), then multiply by the reciprocal of the coefficient on x. Type each move in its box, then check.
Every equation is the same three moves. Rewrite, add the opposite, multiply by the reciprocal. When a move has nothing to do, say so and go on.
Words we use
| Word | Plain English | What you do |
|---|---|---|
| Translate (Step 1) | Rewrite subtraction and division as addition and multiplication, and show the invisible 1s. | Do it first, before any undoing. |
| Equation | A balance that says two sides are equal. | Keep both sides equal at every step. |
| Solve | Find the value of the letter that makes it true. | Get the letter by itself. |
| Coefficient | The number on the variable. It means the variable is multiplied by that number: 3x is 3 times x. | Write it down, even when it is 1. A lonely x is 1x. Anything you leave to do in your head is a top cause of mistakes. |
| Inverse operation (undo partner) | The move that undoes another. Adding undoes adding a negative; multiplying undoes multiplying. | Undo by adding the opposite (makes 0) or multiplying by the reciprocal (makes 1). |
| Do the same to both sides | Keep the balance level. | Whatever you add or multiply on one side, do to the other. |
The big idea
Solving an equation is Lessons 1, 2, and 3 put to work. Step 1 is always translate (Lesson 3): turn subtraction into adding the opposite, division into multiplying by the inverse, and reveal the invisible 1s. Then every line you write is one property (Lesson 2). You add the additive inverse (the opposite) to make the additive identity 0 appear, or you multiply by the multiplicative inverse (the reciprocal) to make the multiplicative identity 1 appear (Lesson 1), and the letter is left standing alone. That is the whole method, and it never changes.
Practice: try it, then get help one step at a time
Try each problem yourself first. Hint gives a nudge, Show next step shows one worked step at a time, Reveal answer checks, and Hide help / start over clears the card. Every problem here needs Step 1 for real: a subtraction to rewrite, a division sign to rewrite, or an invisible 1 to write in. Remember that a fraction like x4 is not a division: it is the coefficient 14 on x. The Step 1 rewrite is shown in this color; the number you add or multiply by in this color.
Exit ticket
On paper or an index card. Use the math word and the plain-English word.
- What is Step 1 of every equation, and why do you do it first?
- Solve 3x - 6 = 9. Show Step 1, then solve and check.
- Why do we do the same thing to both sides? One sentence.
- How is solving an equation the same idea as Lessons 1, 2, and 3? One sentence.
Check your exit ticket
- Step 1 is translate and reveal the invisible 1s: turn subtraction into adding the opposite and division into multiplying by the inverse, and show any hidden 1s. You do it first because the properties and the undo partners only work on addition and multiplication.
- Step 1: 3x - 6 = 3x + (-6). Add 6 to both sides: 3x = 15. Multiply by 13: x = 5. Check: 3 × 5 - 6 = 15 - 6 = 9.
- Any sentence about keeping the balance equal: an equation says two sides are equal, so if you change one side you must change the other the same way.
- Every step is one idea from Lessons 1 to 3: translate first (Lesson 3), then add an opposite to make 0 or multiply by a reciprocal to make 1 (Lessons 1 and 2).
Printables
The worksheet has Parts A to E with an answer key: name the coefficient and what is added on, one-step equations, two-step equations, solve and check, and the exit ticket. The lesson plan has the 25-minute timeline, the vocabulary table, the getting-dressed picture for undo order, and the misconceptions to listen for.