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Math · Variables & One-Step Equations
Variables & One-Step Equations: Inverse Operations, Balance, Solve, Check
Move from numerical expressions to equations by representing unknowns with variables, undoing one operation with its inverse, preserving equality and checking solutions by substitution.
The idea
An equation states that two expressions have the same value. A variable is a symbol, often a letter, that can represent an unknown number.
To solve a one-step equation, identify the operation being done to the variable and use the inverse operation. Addition undoes subtraction, subtraction undoes addition, multiplication undoes division, and division undoes multiplication.
Think of the equals sign as a balance. Whatever valid operation you perform on one side must also be performed on the other side so the equality stays true. Then substitute your answer back into the original equation to check it.
Worked examples
Example 1 · Undo addition
Solve x + 8 = 23.
- The variable has 8 added to it.
- Subtract 8 from both sides: x = 23 − 8.
- x = 15. Check: 15 + 8 = 23.
x = 15.
Example 2 · Undo subtraction
Solve y − 11 = 9.
- The variable has 11 subtracted from it.
- Add 11 to both sides: y = 9 + 11.
- y = 20. Check: 20 − 11 = 9.
y = 20.
Example 3 · Undo multiplication
Solve 6n = 48.
- 6n means 6 × n.
- Divide both sides by 6.
- n = 8. Check: 6 × 8 = 48.
n = 8.
Example 4 · Undo division
Solve p ÷ 5 = 7.
- The variable is divided by 5.
- Multiply both sides by 5.
- p = 35. Check: 35 ÷ 5 = 7.
p = 35.
Example 5 · Translate a context
A number decreased by 14 is 32.
- Let x represent the number.
- 'Decreased by 14' means x − 14.
- Write x − 14 = 32, then add 14 to both sides.
x = 46.
Example 6 · Negative solution
Solve x + 7 = −2.
- Subtract 7 from both sides.
- x = −9.
- Check: −9 + 7 = −2.
x = −9.
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: The equals sign states that two expressions have the same value.
Why this fails: To isolate the variable, undo the operation attached to it.
Why this fails: Unequal changes can destroy the equality.
Why this fails: Undo division with multiplication.
Why this fails: Substitution catches sign, arithmetic and translation errors.
Foundation
Foundation practice
Use inverse operations to solve one-step equations and check by substitution.
1. x + 7 = 19
2. a + 15 = 28
3. y − 8 = 14
4. n − 21 = 9
5. 5p = 35
6. 9k = 72
7. q ÷ 4 = 9
8. r ÷ 6 = 7
9. Does x = 11 solve x + 6 = 17?
10. What does a variable represent in an equation?
Core
Core practice
Solve with whole numbers, decimals and simple fractions.
11. x + 2.5 = 7.1
12. y − 3.7 = 5.2
13. 4n = 18
14. p ÷ 5 = 2.6
15. x + 3/4 = 1 1/2
16. y − 2/5 = 7/10
17. 3x = 2 1/4
18. x ÷ 1/2 = 6
19. Does n = 4.5 solve 4n = 18?
20. Why does doing the same operation to both sides preserve an equation?
Think
Think practice
Translate contexts into one-step equations and analyse common errors.
21. A number plus 18 is 43. Write and solve an equation.
22. After spending $12, Maya has $31 left. How much did she start with?
23. Five equal bags hold 40 marbles. How many in each bag?
24. A 36 km route is split into 4 equal stages. How long is one stage?
25. A student solves x + 9 = 21 by adding 9. Correct the method.
26. A student solves 6x = 42 by subtracting 6. Correct the method.
27. A student solves x ÷ 8 = 5 by dividing by 8 again. Correct it.
28. Which equation matches 'a number decreased by 14 is 20'?
29. Which equation matches 'seven times a number is 56'?
30. What is the first question to ask before solving a one-step equation?
Challenge
Challenge practice
Use negative numbers, unknowns on either side of an equality, and multi-representation reasoning while keeping each equation one-step.
31. x + 9 = −4
32. y − 6 = −15
33. −4n = 28
34. p ÷ (−3) = 8
35. 17 = x + 5
36. 42 = 6n
37. The temperature rose 11°C to reach 4°C. What was the starting temperature?
38. A diver is at −18 m after descending 7 m. What was the starting depth?
39. Without solving exactly, is the solution to 8x = 70 a little less or a little more than 9?
40. Write a one-step equation with solution x = 12.
Answer key
- 1. x = 12
Subtract 7 from both sides: 19 − 7 = 12. - 2. a = 13
Subtract 15 from both sides. - 3. y = 22
Add 8 to both sides. - 4. n = 30
Add 21 to both sides. - 5. p = 7
Divide both sides by 5. - 6. k = 8
Divide both sides by 9. - 7. q = 36
Multiply both sides by 4. - 8. r = 42
Multiply both sides by 6. - 9. Yes
Substitute 11: 11 + 6 = 17. - 10. A number that may be unknown or can vary.
A variable is a symbol, often a letter, used to stand for a number. - 11. x = 4.6
Subtract 2.5 from both sides. - 12. y = 8.9
Add 3.7 to both sides. - 13. n = 4.5
Divide both sides by 4. - 14. p = 13
Multiply both sides by 5. - 15. x = 3/4
Subtract 3/4 from 1 1/2. - 16. y = 1 1/10
Add 2/5 = 4/10 to 7/10. - 17. x = 3/4
Divide 2 1/4 by 3. - 18. x = 3
Multiply both sides by 1/2. - 19. Yes
4 × 4.5 = 18. - 20. It keeps both sides equal.
An equation is a balance; equal changes to both sides preserve equality. - 21. x + 18 = 43; x = 25
Subtract 18 from both sides. - 22. x − 12 = 31; x = 43
Add 12 to both sides. - 23. 5x = 40; x = 8
Divide both sides by 5. - 24. 36 ÷ 4 = x; x = 9 km
The unknown is one equal share. - 25. x = 12
Undo +9 with −9, not +9. - 26. x = 7
Undo multiplication by dividing by 6. - 27. x = 40
Undo division by 8 with multiplication by 8. - 28. x − 14 = 20
Decreased by 14 means subtract 14 from the unknown. - 29. 7x = 56
Seven times a number means multiply the unknown by 7. - 30. What operation is being done to the variable?
Identify the operation so you can choose its inverse. - 31. x = −13
Subtract 9 from both sides. - 32. y = −9
Add 6 to both sides. - 33. n = −7
Divide both sides by −4. - 34. p = −24
Multiply both sides by −3. - 35. x = 12
The sides may be reversed; subtract 5 from both sides. - 36. n = 7
Divide both sides by 6. - 37. x + 11 = 4; x = −7°C
Subtract 11 from 4. - 38. x − 7 = −18; x = −11 m
Add 7 to both sides. - 39. A little less than 9.
8 × 9 = 72, which is slightly above 70. - 40. Answers vary; for example x + 5 = 17.
Any equation that becomes true when x = 12 is acceptable.