Expression Work
Distribute and combine like terms before using inverse operations.
Free Algebra Guide
Multi-Step Equations
Learn how to simplify and solve equations that may require distribution, combining like terms, and several inverse operations before the variable is isolated.
A multi-step equation requires more than two algebra steps to solve. Before isolating the variable, you may need to distribute, combine like terms, simplify constants, or simplify both sides of the equation. The key is to simplify first and solve second.
On This Page
- What makes an equation multi-step
- Why you simplify before solving
- Using the distributive property
- Combining like terms
- Distribution and like terms together
- Using inverse operations afterward
- Negative values
- Fractional coefficients and solutions
- Checking solutions
- Worked examples
- Common mistakes
- A reliable strategy
- Interactive practice
What Is a Multi-Step Equation?
A multi-step equation requires several operations before the variable can be isolated. Some equations need simplification first, while others require multiple inverse operations after simplifying.
3(x + 4) + 2x = 27
Parentheses appear first.
Use the distributive property.
Use the distributive property.
Like terms remain afterward.
Combine the x-terms.
Combine the x-terms.
Then solve the simplified equation.
Undo the constant and coefficient.
Undo the constant and coefficient.
3(x + 4) + 2x = 27 → 3x + 12 + 2x = 27 → 5x + 12 = 27 → 5x = 15 → x = 3
Why Simplification Comes First
In a multi-step equation, trying to isolate the variable too early often makes the work harder. First simplify each side as much as possible without changing the equality.
Equation Work
Once simplified, undo constants and coefficients while keeping both sides balanced.
Simplifying Is Not the Same as Solving
Combining like terms changes the form of one side of the equation. Inverse operations change both sides in a balanced way to isolate the variable.
Using the Distributive Property
When a factor appears outside parentheses, multiply it by every term inside the parentheses before doing anything else with those terms.
Example: 3(x + 4) + 2x = 27
Start
3(x + 4) + 2x = 27
Distribute 3
3x + 12 + 2x = 27
Multiply Every Term Inside
3(x + 4) becomes 3x + 12, not 3x + 4.
Combining Like Terms
After distribution, combine terms with identical variable parts. Constants can also be combined with other constants.
Current equation
3x + 12 + 2x = 27
Combine x-terms
5x + 12 = 27
Keep Unlike Terms Separate
5x and 12 cannot be combined because one is a variable term and the other is a constant.
Distribution and Like Terms in the Same Equation
Many multi-step equations require both skills. Work in a clean order: distribute first, then combine like terms.
Example: −2(x + 4) + 5x = 7
Distribute
−2x − 8 + 5x = 7
Combine like terms
3x − 8 = 7
Using Inverse Operations
Once the equation is simplified, solve it the same way you solve a one-step or two-step equation: remove the constant, then undo the coefficient or divisor.
| Current Equation | Next Step | Result |
|---|---|---|
| 5x + 12 = 27 | Subtract 12 | 5x = 15 |
| 5x = 15 | Divide by 5 | x = 3 |
| 2x − 7 = 21 | Add 7 | 2x = 28 |
| x/4 = 6 | Multiply by 4 | x = 24 |
Apply Inverse Operations to Both Sides
Simplification may happen on one side, but solving operations must preserve equality on both sides.
Multi-Step Equations With Negative Numbers
Negative signs can appear in distribution, coefficients, constants, or solutions. Keep each sign attached to its term throughout the work.
−3(x − 2)
−3x + 6
−4x = 20
x = −5
Multi-Step Equations With Fractions
Fractional coefficients follow the same process. Simplify first, then use the reciprocal or divide as needed to isolate the variable.
Example: 3 4 x + 2x − 5 = 17
Combine like terms
11
4
x − 5 = 17
Remove the constant
11
4
x = 22
Use the reciprocal
x = 8
How to Check a Multi-Step Equation Solution
Substitute the solution into the original equation, not just the simplified version. If both sides have the same value, the solution is correct.
Check x = 3 in 3(x + 4) + 2x = 27
3(3 + 4) + 2(3) = 27
21 + 6 = 27, so 27 = 27 and x = 3 is correct.
Worked Multi-Step Equation Examples
3x + 2x + 4 = 24
- Combine: 5x + 4 = 24.
- Subtract 4: 5x = 20.
- Divide by 5: x = 4.
4(x − 2) + x = 17
- Distribute: 4x − 8 + x = 17.
- Combine: 5x − 8 = 17.
- Add 8, then divide by 5: x = 5.
−2(x + 4) + 5x = 7
- Distribute: −2x − 8 + 5x = 7.
- Combine: 3x − 8 = 7.
- Add 8, then divide by 3: x = 5.
2(x + 3) + 3(x + 4) = 28
- Distribute both groups.
- Combine to get 5x + 18 = 28.
- Subtract 18, then divide by 5: x = 2.
1/2x + 3/2x + 4 = 16
- Combine: 2x + 4 = 16.
- Subtract 4: 2x = 12.
- Divide by 2: x = 6.
1/2(4x + 6) + x = 12
- Distribute: 2x + 3 + x = 12.
- Combine: 3x + 3 = 12.
- Subtract 3, then divide by 3: x = 3.
Common Mistakes With Multi-Step Equations
Solving Before Simplifying
Distribute and combine like terms before trying to isolate the variable.
Distributing to Only One Term
The outside factor must multiply every term inside the parentheses.
Combining Unlike Terms
Variable terms and constants cannot be combined unless their variable parts match.
Losing a Negative Sign
Keep each sign attached to its term, especially during negative distribution.
Changing Only One Side
Inverse operations used to solve must be applied to both sides of the equation.
Skipping Algebra Steps
Write one valid updated equation at a time so errors are easier to catch.
Steps for Solving Multi-Step Equations
1. Distribute if needed
Remove parentheses by multiplying the outside factor by every term inside.
2. Combine like terms
Simplify matching variable terms and constants.
3. Simplify both sides
Make each side as simple as possible before solving.
4. Remove the constant
Use addition or subtraction on both sides.
5. Remove the coefficient or divisor
Use division, multiplication, or a reciprocal to isolate the variable.
6. Write the solution
The variable should now be alone.
7. Check
Substitute the solution into the original equation.
Simplify first. Solve second. Check last.
Practice Solving Multi-Step Equations
Work one algebra step at a time through distribution, like terms, inverse operations, negative values, and fractions.
Open the Multi-Step Equations ToolRelated Math Resources
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