Skill: one-two-step-equations

One- & Two-Step Equations

An equation is a balanced scale: whatever you do to one side, you must do to the other. Inverse operations — done in reverse order — unlock x every time.

1 Understand

The core idea in plain language.

An equation states that two expressions are equal: 2x + 3 = 11. Solving means finding the x that makes it true. The golden rule: do the same thing to both sides. Add 5 to the left? Add 5 to the right. The balance never breaks.

Inverse operations undo each other: addition ↔ subtraction, multiplication ↔ division. In x + 5 = 12, the +5 is undone by subtracting 5 from both sides: x = 7. In 3x = 12, divide both sides by 3: x = 4.

A two-step equation like 2x + 3 = 11 is undone in reverse order of operations: PEMDAS does × before +, so you undo + before ×. Subtract 3 from both sides → 2x = 8; divide by 2 → x = 4.

Always check by substitution: put the answer back into the original equation. If x = 4, then 2(4) + 3 = 11 — true. A check takes ten seconds and catches nearly every slip.

2 See it

Diagrams that make the idea visual.

Undo in reverse: 2x + 3 = 11

PEMDAS built it as “×2, then +3”. Unbuild it backwards: “−3, then ÷2”.

2x + 3 = 11 → 2x = 8 → x = 4.

Undo addition/subtraction first, then multiplication/division.

The balance never breaks

Picture a scale: 2x + 3 on the left pan, 11 on the right. Remove 3 from both pans — it stays level. Halve both pans — still level.

Anything done to only one side tips the scale and breaks the equation.

Same move on both pans: the scale stays level, the equation stays true.

3 The key rule

The solving order

undo +/− first, then ×/÷ — always on both sides

For ax + b = c: subtract/add b from both sides, then divide/multiply by a. For x/a + b = c: clear b first, then multiply by a. Check by substitution.

4 Worked examples

Follow each step. The pattern is always the same.

Example 1 One step: addition

Undo +5 with −5 — on both sides.
x + 5 − 5 = 12 − 5 → x = 7.

x + 5 = 12 → x = 7

Check: 7 + 5 = 12. Correct.

Example 2 One step: division with a negative

Undo ×(−4) with ÷(−4) — on both sides.
x = 20 ÷ (−4) = −5.

−4x = 20 → x = −5

Check: −4(−5) = 20. Correct.

Example 3 Two steps

Reverse order: undo +3 first: 2x = 11 − 3 = 8.
Then undo ×2: x = 8 ÷ 2 = 4.

2x + 3 = 11 → x = 4

Check: 2(4) + 3 = 11. Correct.

Example 4 Two steps with division

Undo +2 first: x/4 = 5 − 2 = 3.
Then undo ÷4 with ×4: x = 3 × 4 = 12.

x/4 + 2 = 5 → x = 12

Check: 12/4 + 2 = 3 + 2 = 5. Correct.

5 Common mistakes

These errors show up on almost every quiz. Spot them now and they will never cost you points.

1. Adding when you should subtract

Wrong

x + 5 = 12 → x = 17 — “I added.”

Right

Use the inverse: subtract 5 → x = 7.

To undo +5, do −5. Inverse operations, not matching ones.

2. Dividing first in a two-step equation

Wrong

2x + 3 = 11 → x + 3 = 5.5 — divided before subtracting.

Right

Reverse order: subtract 3 first → 2x = 8 → x = 4.

Undo in reverse PEMDAS: +/− before ×/÷.

3. Skipping the substitution check

Wrong

“x = 5 looks right” — but 2(5) + 3 = 13, not 11.

Right

Substitute: 2(4) + 3 = 11. Ten seconds, error caught.

The check is part of solving, not extra credit.

6 Key vocabulary

Say these words like you mean them.

Equation: A statement that two expressions are equal. Has an = sign.
Solution: The value of the variable that makes the equation true.
Inverse operations: Pairs that undo each other: +/−, ×/÷.
Isolate: Get the variable alone on one side.
Substitute: Replace the variable with a value to test it.
One-step / two-step: How many inverse operations the solution needs.

7 Quick check

Try each one on paper first, then reveal the answer.

1. Solve: x − 7 = 3.

Answer

Add 7 to both sides: x = 10.

2. Solve: x/5 = 4.

Answer

Multiply both sides by 5: x = 20.

3. Solve: 3x − 4 = 11.

Answer

Add 4: 3x = 15; divide by 3: x = 5.

4. Is x = 6 a solution of 2x + 1 = 13?

Answer

2(6) + 1 = 13 — true. Yes.

Key points to remember

Same move on both sides — the balance rule behind everything.
One-step: apply the inverse operation to both sides.
Two-step: undo in reverse order — +/− first, then ×/÷.
Division form x/a = c: multiply both sides by a.
Check by substitution — every time, no exceptions.

Practice problems Try the tool Explore the lab