1.2Solving Two-Step Equations
Apply two inverse operations in the correct order to isolate the variable. Undo addition or subtraction first, then undo multiplication or division. Always verify your solution.
Why This Matters
Two-step equations are the bridge between basic arithmetic and real algebra. You'll use this exact process to solve distance-rate-time problems in Physics and cost-revenue problems in economics.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
When an equation requires two operations to solve, in what order should you undo them?
Lesson Overview
A two-step equation requires exactly two inverse operations to isolate the variable. The strategy is to reverse the order of operations: since multiplication and division are performed before addition and subtraction in the order of operations, we undo addition or subtraction first, then undo multiplication or division second. Think of it as peeling away the operations that are farthest from the variable first. Every step must be applied to both sides of the equation to keep it balanced.
Worked Examples
Solve: 2x + 5 = 13
Step 1 — Undo addition: subtract 5 from both sides
2x + 5 − 5 = 13 − 5
2x = 8
Step 2 — Undo multiplication: divide both sides by 2
x = 4
Check: 2(4) + 5 = 13 ✓
Solve: 3x − 7 = 11
Step 1 — Undo subtraction: add 7 to both sides
3x = 18
Step 2 — Divide by 3
x = 6
Check: 3(6) − 7 = 11 ✓
Solve: x/4 + 3 = 9
Step 1 — Subtract 3: x/4 = 6
Step 2 — Multiply by 4: x = 24
Check: 24/4 + 3 = 9 ✓
Solve: −5x + 2 = −18
Step 1 — Subtract 2: −5x = −20
Step 2 — Divide by −5: x = 4
Check: −5(4) + 2 = −18 ✓
Solve: x/3 − 4 = 2
Step 1 — Add 4: x/3 = 6
Step 2 — Multiply by 3: x = 18
Check: 18/3 − 4 = 2 ✓
Guided Practice
Guided Practice Video: Solving Two-Step Equations
Watch the guided practice walkthrough for two-step equations, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Solve: 4x + 3 = 19
Hint: Step 1: subtract 3 from both sides. Step 2: divide both sides by 4.
Solve: 5x − 10 = 20
Hint: Step 1: add 10 to both sides. Step 2: divide both sides by 5.
Solve: x/2 + 6 = 14
Hint: Step 1: subtract 6 from both sides. Step 2: multiply both sides by 2.
Solve: −3x + 9 = 0
Hint: Step 1: subtract 9 from both sides. Step 2: divide both sides by −3. Watch the sign.
Solve: 6x − 5 = 31
Hint: Step 1: add 5 to both sides. Step 2: divide both sides by 6.
Key Vocabulary
Two-Step Equation
An equation that requires exactly two inverse operations to solve for the variable.
Example: 2x + 5 = 13 requires subtracting 5, then dividing by 2.
Inverse Operation
The opposite operation used to undo another. Addition ↔ Subtraction; Multiplication ↔ Division.
Example: To undo +5, subtract 5. To undo ×3, divide by 3.
Isolate
To get the variable alone on one side of the equation by undoing all operations applied to it.
Example: In 3x = 12, divide by 3 to isolate x.
Coefficient
The number multiplied by the variable. In 3x, the coefficient is 3.
Example: In −4x + 7 = 3, the coefficient is −4.
Balance
The property that both sides of an equation remain equal when the same operation is applied to both.
Example: If 2x = 8, dividing both sides by 2 keeps it balanced.
Practice Questions
Interactive Practice — 5 Questions
What is the solution to 3x + 4 = 19?
Which operation do you undo FIRST when solving 5x − 8 = 22?
Solve: x/4 + 3 = 9
A student solved −2x + 6 = 14 and got x = 4. Is this correct?
A plumber charges $25 flat plus $40/hour. The bill was $145. How many hours did he work?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Solve: 3x + 4 = 16
Solve: 7x − 5 = 30
Solve: x/3 + 2 = 8
Solve: −4x + 12 = −8
Solve: 5x − 15 = 0
Common Mistakes
Dividing before undoing addition/subtraction — e.g., solving 2x + 6 = 14 by dividing by 2 first.
Undo addition/subtraction first, then undo multiplication/division: subtract 6, then divide by 2.
Applying an operation to only one side of the equation.
Whatever you do to one side, do the exact same thing to the other side.
Forgetting the negative when the coefficient is negative — e.g., −3x = 12 → x = 4.
Divide by −3: x = 12 ÷ (−3) = −4.
Skipping the check step after solving.
Substitute your answer back into the original equation to verify both sides are equal.
Math Tips
Undo addition or subtraction first — these are the operations farthest from the variable.
Whatever you do to one side, do the exact same thing to the other side.
When the coefficient is negative, dividing by a negative flips the sign of the answer.
Always check your solution by substituting it back into the original equation.