How to Solve Multi-Step Equations? (+FREE Worksheet!)
Multi-step equations require two or more inverse operations to isolate the variable. They are the logical extension of one-step equations and are among the most frequently tested skills in Algebra 1. A consistent, organized approach makes even the most complex-looking equations straightforward.
What Is a Multi-Step Equation?
A multi-step equation is any linear equation that cannot be solved in a single operation. For example, \(\color{blue}{2x + 3 = 11}\) requires first subtracting 3 and then dividing by 2. The goal is always the same: isolate the variable on one side of the equation.
How to Solve Multi-Step Equations
Step 1: Simplify each side
Use the distributive property to remove parentheses and combine any like terms on each side first.
- \(\color{blue}{3(x + 2) + x = 14 \rightarrow 3x + 6 + x = 14 \rightarrow 4x + 6 = 14}\)
Step 2: Move all variable terms to one side
Add or subtract terms to collect the variable on one side.
Step 3: Move all constants to the other side
Add or subtract constants so only the variable term remains on its side.
- \(\color{blue}{2x + 3 = 11 \rightarrow 2x = 8}\)
Step 4: Divide (or multiply) to solve for the variable
- \(\color{blue}{2x = 8 \rightarrow x = 4}\)
Step-by-Step Summary
- Distribute and combine like terms on each side.
- Isolate the variable term: add or subtract constants from both sides.
- Solve for the variable: divide or multiply both sides by the coefficient.
- Check your answer by substituting back into the original equation.
Watch: Solving Multi-Step Equations (Video Lesson)
Math with Mr. J demonstrates solving multi-step equations with a clear, organized method:
Multi-Step Equations – Worked Examples
Example 1: Solve \(\color{blue}{2x + 3 = 11}\).
Subtract 3 from both sides: \(\color{blue}{2x = 8}\). Divide by 2: \(\color{blue}{x = 4}\).
Check: \(\color{blue}{2(4) + 3 = 8 + 3 = 11 &\#x2714;}\)
Answer: \(\color{blue}{x = 4}\)
Example 2: Solve \(\color{blue}{3x – 5 = 16}\).
Add 5 to both sides: \(\color{blue}{3x = 21}\). Divide by 3: \(\color{blue}{x = 7}\).
Check: \(\color{blue}{3(7) – 5 = 21 – 5 = 16 &\#x2714;}\)
Answer: \(\color{blue}{x = 7}\)
Example 3: Solve \(\color{blue}{4x + 2 = 18}\).
Subtract 2: \(\color{blue}{4x = 16}\). Divide by 4: \(\color{blue}{x = 4}\).
Check: \(\color{blue}{4(4) + 2 = 18 &\#x2714;}\)
Answer: \(\color{blue}{x = 4}\)
Example 4: Solve \(\color{blue}{5x – 3 = 22}\).
Add 3: \(\color{blue}{5x = 25}\). Divide by 5: \(\color{blue}{x = 5}\).
Check: \(\color{blue}{5(5) – 3 = 25 – 3 = 22 &\#x2714;}\)
Answer: \(\color{blue}{x = 5}\)
More Practice: Multi-Step Equations Video
Khan Academy explains the conceptual reason for keeping both sides balanced — a key insight for all equation solving:
Exercises for Multi-Step Equations
Solve for the variable.
- \(\color{blue}{4x + 4 = 20}\)
- \(\color{blue}{2x – 7 = 15}\)
- \(\color{blue}{3x – 3 = 9}\)
- \(\color{blue}{5x + 5 = 25}\)
- \(\color{blue}{6x – 8 = 10}\)
- \(\color{blue}{4x – 4 = 12}\)
Answers
- \(\color{blue}{x = 4}\)
- \(\color{blue}{x = 11}\)
- \(\color{blue}{x = 4}\)
- \(\color{blue}{x = 4}\)
- \(\color{blue}{x = 3}\)
- \(\color{blue}{x = 4}\)
Frequently Asked Questions
What is the first step in solving a multi-step equation?
Always simplify first: distribute any coefficients through parentheses and combine like terms on each side. Then use inverse operations to isolate the variable.
Does the order of operations matter when solving?
When solving, you reverse the order of operations: undo addition/subtraction first (by doing the opposite), then undo multiplication/division. This is the reverse of PEMDAS.
What if the variable appears on both sides?
Collect all variable terms on one side first by adding or subtracting the variable term from both sides. For example: \(\color{blue}{5x + 2 = 3x + 10 \rightarrow 2x = 8 \rightarrow x = 4}\).
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