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Math · Two-Step Equations

Two-Step Equations: Undo in Reverse Order, Balance, Solve, Check

Build on one-step equations by undoing two operations in reverse order, preserving equality on both sides and checking each solution by substitution.

Grades 5–6 · 40 practice questions · Original material

Learning objectiveInterpret two-step equations, identify the outer operation first, apply inverse operations in reverse order, keep both sides balanced and verify solutions by substitution.

The idea

A two-step equation has two operations connected to the variable. Solving means undoing those operations until the variable stands alone.

Undo operations in reverse order. In 3x + 5 = 23, addition by 5 happened after multiplication by 3, so subtract 5 first and divide by 3 second.

Whatever operation you perform on one side of an equation must also be performed on the other side. A final substitution check catches many sign and order errors.

Worked examples

Example 1 · Multiply, then add

Solve 3x + 5 = 23.

Subtract 5 from both sides: 3x = 18.
Divide both sides by 3: x = 6.
Check: 3(6) + 5 = 18 + 5 = 23.

x = 6.

Example 2 · Multiply, then subtract

Solve 4n − 7 = 21.

Add 7 to both sides: 4n = 28.
Divide both sides by 4: n = 7.
Check: 4(7) − 7 = 28 − 7 = 21.

n = 7.

Example 3 · Divide, then add

Solve y ÷ 5 + 6 = 10.

Subtract 6 from both sides: y ÷ 5 = 4.
Multiply both sides by 5: y = 20.
Check: 20 ÷ 5 + 6 = 4 + 6 = 10.

y = 20.

Example 4 · Divide, then subtract

Solve p ÷ 3 − 4 = 5.

Add 4 to both sides: p ÷ 3 = 9.
Multiply both sides by 3: p = 27.
Check: 27 ÷ 3 − 4 = 9 − 4 = 5.

p = 27.

Example 5 · A negative solution

Solve 2a + 9 = 3.

Subtract 9 from both sides: 2a = −6.
Divide both sides by 2: a = −3.
Check: 2(−3) + 9 = −6 + 9 = 3.

a = −3.

Example 6 · From a situation

Three identical notebooks plus a $4 folder cost $25. Solve 3c + 4 = 25.

Subtract the fixed $4 first: 3c = 21.
Divide by 3: c = 7.
Check: 3($7) + $4 = $25.

Each notebook costs $7.

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Dividing before removing the added number in 3x + 5 = 23.

Why this fails: The +5 is the outer operation. Undo it first; otherwise every term must be divided carefully and the work becomes error-prone.

Changing only one side of the equation.

Why this fails: Equality is preserved only when the same operation is applied to both sides.

Changing a sign without using an operation.

Why this fails: A term does not cross the equals sign and magically change sign; subtract or add the same amount on both sides.

Stopping after the first inverse operation.

Why this fails: After one step, the variable may still be multiplied or divided. Continue until it is isolated.

Skipping the substitution check.

Why this fails: Replacing the variable with your answer verifies that the original left and right sides really match.

Foundation

Foundation practice

Solve two-step equations with positive whole-number solutions.

Two-step equation

1. 2x + 3 = 15

Two-step equation

2. 5a + 4 = 29

Two-step equation

3. 3m − 2 = 19

Two-step equation

4. 6n − 5 = 31

Division equation

5. q ÷ 4 + 2 = 8

Division equation

6. r ÷ 5 − 3 = 4

Two-step equation

7. 7k + 1 = 36

Two-step equation

8. 4t − 9 = 19

Check

9. Does x = 8 solve 3x + 2 = 26?

Order

10. In 5x + 7 = 32, which inverse operation should come first?

Core

Core practice

Mix structures and include zero or negative solutions.

Two-step equation

11. 8x + 6 = 54

Two-step equation

12. 9y − 11 = 34

Negative solution

13. 3a + 8 = 2

Zero solution

14. 4b − 12 = −12

Division equation

15. c ÷ 6 + 5 = 9

Division equation

16. d ÷ 7 − 2 = 6

Two-step equation

17. 2p − 15 = −5

Two-step equation

18. 5q + 12 = 7

Error analysis

19. A student solves 4x + 3 = 27 by dividing 27 by 4 first. What should happen first?

Check

20. Does n = −3 solve 2n + 10 = 4?

Reasoning

Reasoning practice

Translate short situations into equations, then solve and check.

Situation

21. A $5 fee plus 3 equal tickets costs $26. What is each ticket price?

Situation

22. Four equal bags plus 2 loose apples make 30 apples. How many apples are in each bag?

Situation

23. A number is multiplied by 6 and then decreased by 8 to make 40. Find the number.

Situation

24. A number divided by 4, then increased by 3, equals 10. Find the number.

Equation choice

25. Which equation matches: twice a number, plus 9, is 25?

Equation choice

26. Which equation matches: one fifth of a number, minus 2, is 6?

Reasoning

27. If 3x + 4 = 19, what is 3x before solving for x?

Reasoning

28. If y ÷ 8 − 1 = 4, what is y ÷ 8 before solving for y?

Compare methods

29. Why is substitution useful after solving?

Balance

30. If you subtract 6 from the left side of an equation while solving, what must you do to the right side?

Stretch

Stretch practice

Handle larger values, negatives and missing-step reasoning.

Two-step equation

31. 12x + 17 = 101

Two-step equation

32. 15m − 28 = 62

Negative solution

33. 7n + 20 = −1

Negative solution

34. 5p − 6 = −31

Division equation

35. q ÷ 9 + 7 = 12

Division equation

36. r ÷ 12 − 4 = 3

Missing step

37. 4x + 9 = 41 → 4x = 32. What operation produced the second equation?

Missing step

38. 6y − 5 = 43 → 6y = 48. What operation produced the second equation?

Check

39. A student says x = 4 solves 5x + 8 = 33. Is the student correct?

Explain

40. Why do we say to undo operations in reverse order?

Answer key

1. x = 6
Subtract 3: 2x = 12. Divide by 2.
2. a = 5
Subtract 4: 5a = 25. Divide by 5.
3. m = 7
Add 2: 3m = 21. Divide by 3.
4. n = 6
Add 5: 6n = 36. Divide by 6.
5. q = 24
Subtract 2: q ÷ 4 = 6. Multiply by 4.
6. r = 35
Add 3: r ÷ 5 = 7. Multiply by 5.
7. k = 5
Subtract 1: 7k = 35. Divide by 7.
8. t = 7
Add 9: 4t = 28. Divide by 4.
9. Yes
3(8) + 2 = 24 + 2 = 26.
10. Subtract 7 from both sides.
The addition of 7 is the outer operation, so undo it first.
11. x = 6
Subtract 6 to get 8x = 48, then divide by 8.
12. y = 5
Add 11 to get 9y = 45, then divide by 9.
13. a = −2
Subtract 8: 3a = −6. Divide by 3.
14. b = 0
Add 12: 4b = 0. Divide by 4.
15. c = 24
Subtract 5: c ÷ 6 = 4. Multiply by 6.
16. d = 56
Add 2: d ÷ 7 = 8. Multiply by 7.
17. p = 5
Add 15: 2p = 10. Divide by 2.
18. q = −1
Subtract 12: 5q = −5. Divide by 5.
19. Subtract 3 from both sides.
Undo the outer +3 before undoing multiplication by 4.
20. Yes
2(−3) + 10 = −6 + 10 = 4.
21. $7
3t + 5 = 26. Subtract 5, then divide by 3.
22. 7 apples
4b + 2 = 30. Subtract 2, then divide by 4.
23. 8
6x − 8 = 40. Add 8, then divide by 6.
24. 28
x ÷ 4 + 3 = 10. Subtract 3, then multiply by 4.
25. 2x + 9 = 25
Twice a number is 2x; then add 9 and set it equal to 25.
26. x ÷ 5 − 2 = 6
One fifth of x is x ÷ 5; then subtract 2.
27. 15
Subtract 4 from both sides first.
28. 5
Add 1 to both sides first.
29. It checks whether the solution makes the original equation true.
A correct solution gives equal values on both sides of the original equation.
30. Subtract 6 from the right side too.
Applying the same operation to both sides preserves equality.
31. x = 7
Subtract 17: 12x = 84. Divide by 12.
32. m = 6
Add 28: 15m = 90. Divide by 15.
33. n = −3
Subtract 20: 7n = −21. Divide by 7.
34. p = −5
Add 6: 5p = −25. Divide by 5.
35. q = 45
Subtract 7: q ÷ 9 = 5. Multiply by 9.
36. r = 84
Add 4: r ÷ 12 = 7. Multiply by 12.
37. Subtract 9 from both sides.
Subtracting 9 cancels +9 and changes 41 to 32.
38. Add 5 to both sides.
Adding 5 cancels −5 and changes 43 to 48.
39. No; x = 5.
5(4) + 8 = 28, not 33. Solving gives 5x = 25, so x = 5.
40. Because the outermost or most recent operation must be removed before the earlier operation on the variable can be undone cleanly.
For ax + b = c, undo +b first, then undo multiplication by a.

Related practice

Variables & One-Step Equations Expressions & Order of Operations Integers