First Inverse
Remove the constant first: subtraction undoes addition, and addition undoes subtraction.
Free Pre-Algebra Guide
Two-Step Equations
Learn how to isolate a variable using two inverse operations, keep both sides balanced, and check whether the solution makes the original equation true.
A two-step equation requires two inverse operations to isolate the variable. Undo the added or subtracted constant first, then undo the multiplication or division attached to the variable. Every operation must be applied to both sides.
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What Is a Two-Step Equation?
A two-step equation has two operations separating the variable from being alone. For example, in 3x + 5 = 20, x is multiplied by 3 and then 5 is added.
3x + 5 = 20
x is the variable.
It represents the unknown value.
It represents the unknown value.
3x is the variable term.
The variable is multiplied by 3.
The variable is multiplied by 3.
+5 is the constant operation.
This is the outer operation removed first.
This is the outer operation removed first.
The Equal Sign Means “Has the Same Value As”
The equal sign still means both sides have the same value. Solving changes the form of the equation without changing that equality.
Inverse Operations
Inverse operations still undo each other, but in a two-step equation you usually apply them in reverse order: undo addition or subtraction first, then multiplication or division.
Second Inverse
Then isolate the variable: division undoes multiplication, and multiplication undoes division.
| Equation | First Inverse | Second Inverse |
|---|---|---|
| 3x + 5 = 20 | Subtract 5 | Then divide by 3 |
| 4x − 7 = 13 | Add 7 | Then divide by 4 |
| x/3 + 2 = 7 | Subtract 2 | Then multiply by 3 |
| x/5 − 4 = 6 | Add 4 | Then multiply by 5 |
The Balance Rule
An equation remains true when the same operation is applied to both sides.
Example: Solve 3x + 5 = 20
3x + 5 − 5 = 20 − 5
Subtract 5 from both sides first. The constant disappears and the equation becomes 3x = 15. Then divide both sides by 3.
3x = 15 → x = 5
Never Change Only One Side
Whatever operation you apply to one side must also be applied to the other.
Solving Equations Like ax + b = c
Remove the added constant first, then divide both sides by the coefficient multiplying the variable.
Original equation
4x + 7 = 31
Undo the constant
4x + 7 − 7 = 31 − 7
Intermediate equation
4x = 24
Undo the coefficient
4x ÷ 4 = 24 ÷ 4
Solution
x = 6
Solving Equations Like ax − b = c
Add the subtracted constant to both sides first. Then divide by the coefficient to isolate the variable.
Example: Solve 5n − 8 = 27
5n − 8 + 8 = 27 + 8
5n = 35 → n = 7
Undoing the Coefficient
After the constant is removed, many two-step equations reduce to a one-step equation such as 4x = 28. Divide both sides by the coefficient.
Intermediate equation: 4x = 28
4x ÷ 4 = 28 ÷ 4
x = 7
Two-Step Equations With Division
If the variable is divided by a number and a constant is also present, remove the constant first. Then multiply both sides by the denominator.
Example: After removing the constant, solve x 5 = 6
5 ·
x
5
= 5(6)
The factor of 5 cancels the denominator of 5.
x = 30
Two-Step Equations With Negative Numbers
The same two-step process applies when constants, coefficients, or solutions are negative. Keep every sign attached to its number.
x + 9 = 4
x = 4 − 9 = −5
−3x = 18
x = 18 ÷ (−3) = −6
Two-Step Equations With Fractions
A two-step equation can produce a fractional solution. After removing the constant, divide by the coefficient and leave the result as a reduced fraction when it does not divide evenly.
Example: Solve x 10 = − 4 5
- Multiply both sides by 10.
- The denominator of 10 cancels on the left.
- Compute 10 × (−4/5).
x = 10
·
−
4
5
= −8
x = −8
How to Check a Solution
Substitute the proposed solution into the original equation. If both sides have the same value, the solution is correct.
Check x = 5 in 3x + 5 = 20
3(5) + 5 = 20
20 = 20, so x = 5 is correct.
Worked Two-Step Equation Examples
Solve 2x + 7 = 21
- Subtract 7: 2x = 14.
- Divide by 2: x = 7.
Solve 7n − 6 = 29
- Add 6: 7n = 35.
- Divide by 7: n = 5.
Solve 4y + 3 = −17
- Subtract 3: 4y = −20.
- Divide by 4: y = −5.
Solve −3m + 4 = 19
- Subtract 4: −3m = 15.
- Divide by −3: m = −5.
Solve p/5 − 3 = 4
- Add 3: p/5 = 7.
- Multiply by 5: p = 35.
Solve 8a + 2 = 7
- Subtract 2: 8a = 5.
- Divide by 8: a = 5/8.
Common Mistakes With Two-Step Equations
Undoing the Coefficient First
In 3x + 5 = 20, remove the +5 before dividing by 3.
Changing Only One Side
Apply the same operation to both sides.
Ignoring a Negative Coefficient
In −4x = 20, divide by −4, not 4.
Forgetting to Work in Reverse Order
Undo the outside addition or subtraction before the multiplication or division.
Forcing a Fraction Into a Whole Number
If the final division does not come out evenly, a reduced fraction may be the correct answer.
Skipping the Check
Substitution catches sign and arithmetic errors.
Steps for Solving Two-Step Equations
1. Identify the constant operation
Find the addition or subtraction outside the variable term.
2. Undo the constant
Add or subtract the same value on both sides.
3. Simplify
Write the intermediate one-step equation.
4. Undo the coefficient or divisor
Divide or multiply both sides as needed.
5. Write the solution
The variable should now be isolated.
6. Check
Substitute into the original equation.
Practice Solving Two-Step Equations
Undo the constant first, simplify the intermediate equation, undo the coefficient or divisor, and verify your result.
Open the Two-Step Equations ToolRelated Math Resources
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