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State Board · Class 8 · Mathematics

Lesson 12: Equations in One Variable

3 concepts · 24 practice questions with answers · Free to read

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An equation is a balance

From daily life

A balance has x + 7 in one pan and 15 in the other, and it is level. Take 7 off both pans and it stays level: x = 8.

Simple definition

A statement saying two sides are equal is an equation. The value of the letter that makes it true is the solution.

x + 715− 7− 7x = 8

In the figure: x + 7 = 15: take 7 off both pans and the balance stays level ⇒ x = 8.

Formula: Do the same thing to both sides: + , − , × , ÷ (not by 0)

Live figure

An equation is a balance

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Which operation to use

Operation on xDo to both sides
x + 5subtract 5
x − 5add 5
5xdivide by 5
x ÷ 5multiply by 5
  • Easy: x − 6 = 9 ; add 6 to both sides ; x = 15.
  • Medium: 3x = 36 ; divide both sides by 3 ; x = 12.
  • Exam-type: x ÷ 4 = 5 ; multiply both sides by 4 ; x = 20.
Remember

Check: put the value in the original equation and see that both sides match.

Common mistake

Operating on one side only. Whatever you do on one side, do on the other.

Shortcut

Undo what has been done to x: − for +, ÷ for ×. Memory clue: Balance: whatever you do on one pan, do on the other pan too.

Spot the difference

PointEquationAlgebraic expression
Sign =presentabsent
Meaningtwo sides are equaljust one expression
Examplex + 7 = 15x + 7
Can it be solvedyes (x = 8)no, only simplified

Practice questions

Easy
  1. 1x + 9 = 20
  2. 24x = 52
  3. 3x ÷ 5 = 7
Medium
  1. 4x − 8 = 13
  2. 56x = 84
  3. 6x ÷ 3 + 2 = 9
Challenging
  1. 72x + 5 = 17
  2. 83x − 7 = 20
Show answers (try them yourself first)
  1. 1x = 20 − 9 = 11.
  2. 2x = 52 ÷ 4 = 13.
  3. 3x = 7 × 5 = 35.
  4. 4x = 13 + 8 = 21.
  5. 5x = 84 ÷ 6 = 14.
  6. 6x ÷ 3 = 7; x = 7 × 3 = 21.
  7. 72x = 12; x = 6.
  8. 83x = 27; x = 9. Check: 3 × 9 − 7 = 20 ✔.

Equations with x on both sides

From daily life

Amit has 5 bags of balls but 3 fewer; Sumit has 2 bags and 9 extra. They have the same number: 5x − 3 = 2x + 9 → x = 4 balls per bag.

Simple definition

First open brackets; if there are fractions, multiply every term by the LCM. Then collect x on one side and numbers on the other.

5x − 3 = 2x + 93x − 3 = 9− 2x− 2x3x = 12+ 3+ 3x = 4÷ 3÷ 3

In the figure: 5x − 3 = 2x + 9: subtract 2x from both sides, add 3 to both sides, then divide by 3 ⇒ x = 4.

In words: Steps: 1) Open brackets. 2) Clear fractions. 3) x terms on one side, numbers on the other. 4) Divide by the number with x. 5) Check.

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Equations with x on both sides

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  • Easy: 5x − 3 = 2x + 9 ; 5x − 2x = 9 + 3 ; 3x = 12 ; x = 4. Check: 5 × 4 − 3 = 17 and 2 × 4 + 9 = 17.
  • Medium: 3(x − 2) = 2(x + 3) ; 3x − 6 = 2x + 6 ; 3x − 2x = 6 + 6 ; x = 12. Check: 3 × 10 = 30 and 2 × 15 = 30.
  • Exam-type: x ÷ 2 + x ÷ 3 = 10 ; multiply all by 6: 3x + 2x = 60 ; 5x = 60 ; x = 12. Check: 6 + 4 = 10.
  • Exam-type: (2x + 1) ÷ 3 = 5 ; 2x + 1 = 15 ; 2x = 14 ; x = 7.
Remember

First open brackets, then clear fractions, then bring x to one side.

Common mistake

Multiply the outside number with every term inside: 3(x − 2) = 3x − 6, not just 3x.

Shortcut

When a term crosses to the other side its sign flips: +9 becomes −9, ×3 becomes ÷3. Memory clue: x on one side, numbers on the other: "x here, numbers there".

Spot the difference

Pointx on one sidex on both sides
Examplex − 6 = 95x − 3 = 2x + 9
First stepmove the numberbring the x terms to one side
Number of steps1 to 23 to 4
Answerx = 15x = 4

Practice questions

Easy
  1. 16x + 4 = 3x + 19
  2. 29x − 4 = 5x + 12
  3. 33(x + 4) = 27
Medium
  1. 45(x − 2) = 3(x + 4)
  2. 5x ÷ 2 + x ÷ 5 = 14
  3. 6(3x − 2) ÷ 4 = 7
Challenging
  1. 74(2x − 1) = 3(x + 6) − 2
  2. 8(x + 1) ÷ 2 = (x − 1) ÷ 3 + 2
Show answers (try them yourself first)
  1. 16x − 3x = 19 − 4; 3x = 15; x = 5.
  2. 24x = 16; x = 4.
  3. 33x + 12 = 27; 3x = 15; x = 5.
  4. 45x − 10 = 3x + 12; 2x = 22; x = 11.
  5. 5Multiply by 10: 5x + 2x = 140; 7x = 140; x = 20.
  6. 63x − 2 = 28; 3x = 30; x = 10.
  7. 78x − 4 = 3x + 18 − 2; 8x − 4 = 3x + 16; 5x = 20; x = 4.
  8. 8Multiply by 6: 3(x + 1) = 2(x − 1) + 12; 3x + 3 = 2x + 10; x = 7. Check: 8 ÷ 2 = 4 and 6 ÷ 3 + 2 = 4 ✔.

Making equations from word problems

From daily life

"My father is 3 times my age; in 12 years he will be twice my age." Call the unknown x and write an equation.

Simple definition

Call the unknown x, turn the words into an equation, solve it, then write the answer in words.

x123x36x + 12243x + 1248+ 1248 = 2 × 24

In the figure: Son x = 12, father 3x = 36; after 12 years 24 and 48: 48 = 2 × 24.

Live figure

Making equations from word problems

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In words: Steps: 1) Write what x stands for. 2) Form the equation. 3) Solve. 4) Check against the problem.

  • Easy: A number plus 12 is 30. x + 12 = 30 ; x = 18.
  • Medium: The sum of three consecutive numbers is 63. x + (x + 1) + (x + 2) = 63 ; 3x + 3 = 63 ; 3x = 60 ; x = 20. The numbers are 20, 21, 22.
  • Exam-type: Son is x years, father 3x. After 12 years: 3x + 12 = 2(x + 12) ; 3x + 12 = 2x + 24 ; x = 12. Son 12, father 36. Check: after 12 years 24 and 48.
  • Exam-type: A rectangle is 4 cm longer than wide, perimeter 40. 2(x + x + 4) = 40 ; 4x + 8 = 40 ; x = 8. Width 8, length 12.

Words to maths

WordsMaths
twice a number2x
5 more than a numberx + 5
5 less than a numberx − 5
half of a numberx ÷ 2
Remember

When forming an equation from words, first write down what x stands for.

Common mistake

"5 less than x" is x − 5, but "x subtracted from 5" is 5 − x. They are different.

Shortcut

Check the answer against the conditions of the original problem. Memory clue: In word problems first write "what does x mean?", then the equation.

Spot the difference

Point5 less thanSubtract from 5
Sentence5 less than a numbera number subtracted from 5
Formx − 55 − x
For x = 83−3
Orderchanging it changes the valuechanging it changes the value

Practice questions

Easy
  1. 1x + 9 = 20
  2. 24x = 52
  3. 3x ÷ 5 = 7
Medium
  1. 47x − 5 = 3x + 11
  2. 52(x + 4) = 18
  3. 6(3x − 1) ÷ 2 = 7
Challenging
  1. 7x ÷ 3 + x ÷ 4 = 14
  2. 8The sum of three consecutive even numbers is 54. Find them.
Show answers (try them yourself first)
  1. 1x = 20 − 9 = 11.
  2. 2x = 52 ÷ 4 = 13.
  3. 3x = 7 × 5 = 35.
  4. 44x = 16 ; x = 4. Check: 28 − 5 = 23 and 12 + 11 = 23.
  5. 5x + 4 = 9 ; x = 5.
  6. 63x − 1 = 14 ; 3x = 15 ; x = 5.
  7. 7Multiply all by 12: 4x + 3x = 168 ; 7x = 168 ; x = 24. Check: 8 + 6 = 14.
  8. 8x + (x + 2) + (x + 4) = 54 ; 3x + 6 = 54 ; x = 16 ; the numbers are 16, 18, 20.