Solving Linear Equations Quizzes

Introduction

Solving linear equations is a core skill in GCSE Maths. A linear equation is an equation in which the highest power of the variable is 1. Solving linear equations involves finding the value of the variable that makes the equation true. Mastering linear equations is essential for algebra, problem-solving, and real-life applications such as calculating costs, distances, or quantities.

For example, in the equation $$2x + 5 = 13$$, solving for x gives $$x = 4$$. Understanding the process of solving linear equations allows students to handle more complex algebraic expressions, inequalities, and systems of equations confidently.

Core Concepts

Definition of a Linear Equation

A linear equation is an equation of the form:

$$ax + b = c$$

Where:

• x is the variable
• a, b, c are constants
• a ≠ 0

Basic Principles for Solving

To solve linear equations:

1. Isolate the variable on one side of the equation
2. Perform inverse operations to simplify
3. Maintain balance: whatever you do to one side, do to the other

Inverse Operations

• Addition ↔ Subtraction
• Multiplication ↔ Division
• Used to move terms and isolate the variable

Solving One-Step Equations

Equations that require only one operation to solve:

Examples:

• $$x + 5 = 12 \Rightarrow x = 12 - 5 = 7$$
• $$3x = 15 \Rightarrow x = 15 ÷ 3 = 5$$
• $$x - 4 = 9 \Rightarrow x = 9 + 4 = 13$$

Solving Two-Step Equations

Equations that require two inverse operations:

Examples:

• $$2x + 3 = 11$$
• Step 1: Subtract 3: $$2x = 8$$
• Step 2: Divide by 2: $$x = 4$$
• $$5x - 7 = 18$$
• Step 1: Add 7: $$5x = 25$$
• Step 2: Divide by 5: $$x = 5$$

Equations with Variables on Both Sides

Sometimes, the variable appears on both sides:

Example:

• $$3x + 5 = 2x + 9$$
• Step 1: Subtract 2x from both sides: $$x + 5 = 9$$
• Step 2: Subtract 5 from both sides: $$x = 4$$

Equations with Brackets

When equations contain brackets, expand first, then solve:

Example:

• $$2(x + 3) = 14$$
• Step 1: Expand: $$2x + 6 = 14$$
• Step 2: Subtract 6: $$2x = 8$$
• Step 3: Divide by 2: $$x = 4$$

Equations with Fractions

To solve equations with fractions:

• Multiply both sides by the denominator
• Then solve as usual

Example:

• $$\frac{x}{3} + 2 = 5$$
• Step 1: Subtract 2: $$\frac{x}{3} = 3$$
• Step 2: Multiply by 3: $$x = 9$$

Checking Solutions

Always substitute the solution back into the original equation to verify:

Example:

• Equation: $$2x + 3 = 11$$, solution: $$x = 4$$
• Check: 2 × 4 + 3 = 8 + 3 = 11 → Correct

Word Problems

Linear equations often appear in real-life problems:

• Example: A number plus 7 equals 15. Find the number.
• Equation: $$x + 7 = 15$$
• Subtract 7: $$x = 8$$
• Example: Twice a number minus 5 equals 13. Find the number.
• Equation: $$2x - 5 = 13$$
• Add 5: $$2x = 18$$
• Divide by 2: $$x = 9$$

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Worked Examples

Example 1 (Foundation): One-step equation

$$x + 6 = 12$$

• Subtract 6: $$x = 12 - 6 = 6$$

Example 2 (Foundation): One-step with multiplication

$$4x = 20$$

• Divide by 4: $$x = 5$$

Example 3 (Higher): Two-step equation

$$3x + 5 = 14$$

• Subtract 5: $$3x = 9$$
• Divide by 3: $$x = 3$$

Example 4 (Higher): Variables on both sides

$$5x + 3 = 2x + 12$$

• Subtract 2x: $$3x + 3 = 12$$
• Subtract 3: $$3x = 9$$
• Divide by 3: $$x = 3$$

Example 5 (Higher): Equation with brackets

$$2(x + 4) = 14$$

• Expand: $$2x + 8 = 14$$
• Subtract 8: $$2x = 6$$
• Divide by 2: $$x = 3$$

Example 6 (Higher): Equation with fractions

$$\frac{x}{5} + 3 = 7$$

• Subtract 3: $$\frac{x}{5} = 4$$
• Multiply by 5: $$x = 20$$

Example 7 (Real-life): Word problem

A number minus 8 equals 15. Find the number.

• Equation: $$x - 8 = 15$$
• Add 8: $$x = 23$$

Example 8 (Real-life): Money problem

Twice a number plus 5 equals 17. Find the number.

• Equation: $$2x + 5 = 17$$
• Subtract 5: $$2x = 12$$
• Divide by 2: $$x = 6$$

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Common Mistakes

• Not performing the same operation on both sides
• Forgetting to subtract or add constants correctly
• Dividing by the wrong coefficient
• Not expanding brackets before solving
• Ignoring negative signs

Tips to avoid errors:

• Always isolate the variable on one side
• Follow inverse operations carefully
• Use BODMAS when expanding brackets or handling fractions
• Check solutions by substitution
• Practice real-life and word problems to apply understanding

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Applications

• Solving everyday problems: distances, money, quantities
• Algebra: solving for unknown variables in formulas
• Physics: using linear equations to calculate speed, time, force
• Finance: simple interest, budgeting, cost calculations
• Mathematical reasoning