Simple Equations — Class 7 IMO

Forming Equations

An equation turns a word statement into maths using a variable. 'Five more than a number is 12' becomes x + 5 = 12; 'thrice a number is 21' becomes 3x = 21.

Example 1: Write 'thrice a number is 21' as an equation.

3x = 21.

Example 2: Write 'a number decreased by 4 is 10' as an equation.

x − 4 = 10.

Quick recap

Choose a variable for the unknown.
Translate the words into an equation.

✓ Quick check

Write 'twice a number plus 3 is 15' as an equation.

Twice a number is 2x, plus 3 equals 15.

Write 'a number divided by 5 is 4' as an equation.

A number divided by 5 is x/5 = 4.

Solving by Balancing & Transposing

Two-step equations require two inverse operations to isolate the variable. The general form is ax + b = c.

Steps to solve:

Add or subtract to move the constant term to the other side
Multiply or divide to isolate the variable
Check by substituting your answer back into the original equation

Example: 2x + 3 = 11

Step 1: Subtract 3 from both sides: 2x = 8

Step 2: Divide both sides by 2: x = 4

Key rule: Whatever you do to one side, you must do to the other side.

EQUATION BALANCE SCALE:

   2x + 3  =  11
      |
  Subtract 3 from both sides
      |
    2x    =   8
      |
  Divide both sides by 2
      |
     x    =   4

VERIFY: 2(4) + 3 = 8 + 3 = 11 [correct!]

Example 1: Solve x + 7 = 12.

x = 12 − 7 = 5.

Example 2: Solve 3x = 18.

x = 18 ÷ 3 = 6.

Example 3: Solve: 3x - 7 = 14

3x - 7 + 7 = 14 + 7 => 3x = 21 => x = 7. Check: 3(7)-7 = 14. Correct!

Example 4: Solve: x/4 + 2 = 10

x/4 + 2 - 2 = 10 - 2 => x/4 = 8 => x = 32. Check: 32/4+2 = 10. Correct!

Example 5: Solve: -2x + 5 = 17

-2x + 5 - 5 = 17 - 5 => -2x = 12 => x = -6. Check: -2(-6)+5 = 17. Correct!

Quick recap

Balancing: do the same to both sides.
Transposing: move a term across and change its sign.
Always perform the same operation on BOTH sides
Undo addition/subtraction first, then multiplication/division
Check your answer by plugging it back into the original equation
Negative coefficients: dividing by a negative flips nothing (unlike inequalities)

✓ Quick check

Solve: 2x − 3 = 7.

2x = 10, so x = 5.

Solve: x/4 = 3.

x = 3 × 4 = 12.

Applications

Equations solve real problems about age, numbers, money and geometry. Read the problem, choose a variable, form the equation and solve.

Example 1: A father is three times as old as his son, who is 12. How old is the father?

3 × 12 = 36 years.

Example 2: When 5 is added to a number the result is 20. Find the number.

x + 5 = 20, so x = 15.

Quick recap

Set a variable, write the equation, then solve.
Works for age, number, money and geometry.

✓ Quick check

Solve: 5x = 45.

x = 45 ÷ 5 = 9.

Solve: 2x + 4 = 20.

2x = 16, so x = 8.