Practice Maths

Solving Linear Equations I

Solving Equations by Balancing

Key Terms

equation
a mathematical statement that two expressions are equal
balance
the condition where both sides of an equation are equal
isolate
to get the variable by itself on one side of the equation
inverse operation
the opposite operation that undoes another (e.g. + is undone by −)

Solving Equations by Balancing

An equation is like a balanced scale. The left side equals the right side.

To find the unknown value (the variable), we must isolate it (get it by itself).

Hot Tip Whatever you do to one side of the equation, you MUST do to the other side to keep it balanced.

Example: 2y + 4 = 14

  1. Subtract 4 from both sides:
    2y = 10
  2. Divide by 2 on both sides:
    y = 5

Worked Example

Problem: Solve for x: x + 7 = 15

Step 1 — Identify the operation applied to x.
The equation shows x with 7 added to it. The inverse of adding 7 is subtracting 7.

Step 2 — Apply the inverse operation to both sides.
x + 7 − 7 = 15 − 7
x = 8

Step 3 — Check the answer by substituting back.
8 + 7 = 15. ✓

The Balance Scale: Why It Works

Picture a perfectly balanced scale at a market. On the left: a mystery bag of apples plus 3 loose apples. On the right: 15 loose apples. The scale is balanced, which means both sides weigh the same — they're equal.

Now here's the key insight: whatever you do to one side, you must do to the other side to keep it balanced. If you take 3 apples off the left, you must also take 3 off the right. The scale tips if you don't. This is the entire logic of equation-solving!

The Language of Inverse Operations

Every mathematical operation has an opposite (inverse) that "undoes" it:

  • Addition (+7) is undone by subtraction (−7)
  • Subtraction (−12) is undone by addition (+12)
  • Multiplication (×4) is undone by division (÷4)
  • Division (÷5) is undone by multiplication (×5)

When we solve an equation, we apply inverse operations to isolate the variable — get it alone on one side of the scale.

One-Step Equations: The Simplest Case

Start with x + 7 = 15. The variable x has had 7 added to it. To undo that, subtract 7 from both sides:

  • x + 7 − 7 = 15 − 7
  • x = 8

Now check: 8 + 7 = 15. Correct! Always substitute your answer back into the original equation to verify it works.

Two-Step Equations: Peel the Layers

Think of solving 2y + 4 = 14 like peeling an onion — remove the outer layer first, then the inner layer. The outer operation (furthest from x) gets undone first:

  • Step 1 — Remove +4 by subtracting 4 from both sides: 2y = 10
  • Step 2 — Remove ×2 by dividing both sides by 2: y = 5
  • Check: 2(5) + 4 = 10 + 4 = 14. Correct!
Remember: Undo operations in REVERSE order to how they were applied. BODMAS says "multiply before add", so when undoing, undo the addition first, then the multiplication. It's reverse BODMAS.

Real-World Application: The Mystery Number

Here's a classic puzzle: "I think of a number, multiply it by 3, then add 2. I get 20. What is my number?" Turn this into algebra:

  • Let the number be n. The equation is: 3n + 2 = 20
  • Subtract 2: 3n = 18
  • Divide by 3: n = 6
  • Check: 3(6) + 2 = 18 + 2 = 20. Correct!
Common Mistake: Students sometimes subtract the multiplier instead of dividing by it. If you see 3n = 18, you need to divide by 3, not subtract 3. Subtracting 3 would give 3n − 3 = 15, which is wrong. Always divide to remove a coefficient.

Practice Questions

  1. Balancing Scales Fluency

    Find the missing value to make each equation balance.

    1. 10 + 5 = ? + 3
    2. 18 − 4 = 7 + ?
    3. 3 × 4 = x + 2
    4. 20 ÷ 2 = y − 5

  2. Inverse Operations Fluency

    Write the inverse (opposite) operation for each of the following.

    1. Adding 7 (+7)
    2. Subtracting 12 (−12)
    3. Multiplying by 4 (× 4)
    4. Dividing by 5 (÷ 5)

  3. Solving One-Step Equations (Addition and Subtraction) Fluency

    Find the value of the pronumeral.

    1. x + 9 = 20
    2. m − 5 = 11
    3. 15 = p + 3

  4. Solving One-Step Equations (Multiplication and Division) Fluency

    Find the value of the pronumeral.

    1. 5y = 35
    2. 3k = 27
    3. t ÷ 2 = 6

  5. Bags of Marbles Understanding

    Imagine a balance scale. On the left side: 2 bags of marbles and 3 loose marbles. On the right side: 15 loose marbles. The scale is balanced.

    1. Write this as an equation (let b = number of marbles in one bag).
    2. Subtract 3 loose marbles from both sides. What is the new equation?
    3. How many marbles are in one bag?

  6. Boxes and Blocks Understanding

    On the left side of a scale: 3 boxes and 2 blocks. On the right side: 14 blocks. The scale is balanced.

    1. Write the equation (let x = number of blocks in one box).
    2. Subtract 2 from both sides. What is the new equation?
    3. How many blocks are in one box?

  7. Solving Two-Step Equations (Guided) Understanding

    Solve for x:   2x + 3 = 13

    1. First, subtract 3 from both sides. What is the new equation?
    2. Now divide both sides by 2. What is x?

  8. Solving Two-Step Equations Understanding

    Use the balance method to solve each equation. Show your working.

    1. 3m − 4 = 11
    2. 5k + 2 = 22
    3. 4p + 10 = 30

  9. Working Backwards Problem Solving

    I think of a number. I multiply it by 3, then add 2. The result is 20.

    1. Write this as an equation (use n for the number).
    2. Solve the equation to find the number.

  10. Solving from Context Problem Solving

    A bus fare costs a $2.50 base charge plus $1.50 per kilometre.

    1. Write the formula for the total fare (C) in terms of kilometres (k).
    2. A passenger is charged $11.50. Write an equation to find the number of kilometres travelled.
    3. Solve the equation. How far was the trip?
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