1.3Solving Multi-Step Equations
Combine like terms, apply the distributive property, and use inverse operations to solve equations that require three or more steps. Build a systematic approach that works every time.
Why This Matters
Multi-step equations are the standard form of real-world problems — almost nothing in science or engineering solves in just one step. Mastering this skill prepares you for Geometry proofs, Physics kinematics, and Precalculus.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
What systematic strategy can you use to solve any equation, no matter how many steps it requires?
Lesson Overview
A multi-step equation requires three or more operations to isolate the variable. The key is a consistent four-step strategy: (1) Distribute to remove parentheses, (2) Combine like terms on each side, (3) Move variable terms to one side, and (4) Isolate the variable using multiplication or division. Not every equation needs all four steps, but applying them in order ensures you never miss anything. Always check your solution.
Worked Examples
Solve: 3x + 5 + 2x = 20
Combine like terms: 5x + 5 = 20
Subtract 5: 5x = 15
Divide by 5: x = 3
Check: 3(3)+5+2(3)=20 ✓
Solve: 2(x + 4) = 18
Distribute: 2x + 8 = 18
Subtract 8: 2x = 10
Divide by 2: x = 5
Check: 2(9)=18 ✓
Solve: 5x − 3 = 2x + 12
Subtract 2x: 3x − 3 = 12
Add 3: 3x = 15
Divide by 3: x = 5
Check: 5(5)−3=22; 2(5)+12=22 ✓
Solve: 3(2x − 1) + 4 = 25
Distribute: 6x − 3 + 4 = 25
Combine: 6x + 1 = 25
Subtract 1: 6x = 24
Divide by 6: x = 4
Check: 3(7)+4=25 ✓
Solve: 4x + 7 = 2(x + 5) + 3
Distribute right: 4x+7=2x+13
Subtract 2x: 2x+7=13
Subtract 7: 2x=6
Divide by 2: x=3
Check: 4(3)+7=19; 2(8)+3=19 ✓
Guided Practice
Guided Practice Video: Solving Multi-Step Equations
Watch the guided practice walkthrough for multi-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 + 2x = 27
Hint: Combine 4x + 2x = 6x. Then solve 6x + 3 = 27.
Solve: 3(x + 5) = 24
Hint: Distribute to get 3x + 15 = 24. Then subtract 15 and divide by 3.
Solve: 7x − 4 = 3x + 16
Hint: Subtract 3x from both sides. You'll get 4x − 4 = 16.
Solve: 2(3x − 2) + 5 = 21
Hint: Distribute to get 6x − 4 + 5. Combine −4 + 5 = 1.
Solve: 5x + 8 = 2(x + 7) + 1
Hint: Distribute the right side. Combine constants. Move variable terms.
Key Vocabulary
Multi-Step Equation
An equation that requires three or more operations to solve for the variable.
Example: 3(2x − 1) + 4 = 25 requires distributing, combining, then isolating.
Like Terms
Terms that have the same variable raised to the same power.
Example: 3x and 7x are like terms; 3x and 3x² are not.
Combine Like Terms
Add or subtract like terms to simplify an expression.
Example: 3x + 7x = 10x
Distributive Property
a(b + c) = ab + ac. Multiply the factor outside by each term inside.
Example: 3(x + 4) = 3x + 12
Practice Questions
Interactive Practice — 5 Questions
Solve: 3x + 5 + 2x = 20
Solve: 2(x + 4) = 18
What is the FIRST step when solving 3(2x − 1) + 4 = 25?
Solve: 5x − 3 = 2x + 12
Solve: 4x + 7 = 2(x + 5) + 3
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Solve: 5x + 2 + 3x = 34
Solve: 2(x + 6) = 26
Solve: 6x − 5 = 4x + 11
Solve: 3(2x + 1) − 4 = 23
Solve: 4x + 9 = 2(x + 8) − 3
Common Mistakes
Forgetting to combine like terms before isolating the variable.
Simplify each side completely first, then isolate the variable.
Distributing to only the first term inside parentheses — e.g., 3(x + 4) = 3x + 4.
Distribute to every term: 3(x + 4) = 3x + 12.
Combining unlike terms — e.g., adding 3x and 5 to get 8x.
Only combine terms with the same variable and exponent. 3x + 5 cannot be simplified further.
Losing track of negative signs when moving terms across the equals sign.
Use inverse operations carefully. Subtract a term from both sides with its sign.
Math Tips
Always distribute before combining like terms — parentheses must be cleared first.
Combine like terms on each side separately before moving terms across the equals sign.
A negative sign in front of parentheses means multiply every term inside by −1.
If the variable cancels and you get a true statement (5 = 5), there are infinitely many solutions.