# IGCSE Mathematics Foundation: Linear Equations

## Scheme of work: IGCSE Foundation: Year 10: Term 4: Linear Equations

#### Prerequisite Knowledge

1. Simplifying Expressions by Collecting Like Terms
• Concept: Understanding how to combine like terms in an algebraic expression to simplify it.
• Example: $\text{Simplify } 3x + 4x – 2x.$ $\text{Combine like terms: } (3 + 4 – 2)x = 5x.$
2. Expanding Expressions
• Concept: Knowing how to expand algebraic expressions, particularly those involving brackets.
• Example: $\text{Expand } 2(x + 3).$ $\text{Distribute the } 2: 2x + 6.$
3. Factorising Expressions
• Concept: Understanding how to factorise algebraic expressions by finding common factors.
• Example: $\text{Factorise } 4x + 8.$ $\text{Find the common factor: } 4(x + 2).$

#### Success Criteria

1. Solve Linear Equations with Integer or Fractional Coefficients in One Unknown, Where the Unknown Appears on Either Side or Both Sides of the Equation
• Objective: Accurately solve linear equations with integer or fractional coefficients, ensuring the unknown is isolated on one side of the equation.
• Example: $\text{Solve } 3x + 5 = 2x + 7.$ $\text{Subtract } 2x \text{ from both sides: } x + 5 = 7.$ $\text{Subtract 5 from both sides: } x = 2.$
• Example with Fractions: $\text{Solve } \frac{2}{3}x – \frac{1}{2} = \frac{1}{6}x + \frac{5}{3}.$ $\text{Multiply through by 6 to clear fractions: } 4x – 3 = x + 10.$ $\text{Subtract } x \text{ from both sides: } 3x – 3 = 10.$ $\text{Add 3 to both sides: } 3x = 13 \rightarrow x = \frac{13}{3}.$
2. Set Up Simple Linear Equations from Given Data
• Objective: Formulate linear equations based on word problems or given data, ensuring accurate representation of the problem context.
• Example: $\text{A number decreased by 7 is equal to twice the number increased by 3. Find the number.}$ $\text{Let the number be } x.$ $x – 7 = 2x + 3.$ $\text{Subtract } x \text{ from both sides: } -7 = x + 3.$ $\text{Subtract 3 from both sides: } -10 = x \rightarrow x = -10.$

#### Key Concepts

1. Solve Linear Equations with Integer or Fractional Coefficients in One Unknown, Where the Unknown Appears on Either Side or Both Sides of the Equation
• Concept: Isolating the variable on one side of the equation is key to solving linear equations. This often involves inverse operations such as addition, subtraction, multiplication, and division to simplify the equation step-by-step.
2. Set Up Simple Linear Equations from Given Data
• Concept: Translating a word problem or given data into an algebraic equation requires identifying key information and relationships. It’s essential to represent the problem context accurately in the equation.

#### Common Misconceptions

1. Solve Linear Equations with Integer or Fractional Coefficients in One Unknown, Where the Unknown Appears on Either Side or Both Sides of the Equation
• Common Mistake: Failing to correctly isolate the variable, often by not performing the same operation on both sides of the equation.
• Example: $\text{Incorrect: Solving } 3x + 5 = 2x + 7 \text{ by subtracting 5 from one side only}.$ $\text{Correct: Subtracting } 2x \text{ from both sides: } x + 5 = 7.$ $\text{Then subtracting 5 from both sides: } x = 2.$
• Common Mistake with Fractions: $\text{Incorrect: Solving } \frac{2}{3}x – \frac{1}{2} = \frac{1}{6}x + \frac{5}{3} \text{ by not clearing fractions correctly}.$ $\text{Correct: Multiply through by 6 to clear fractions: } 4x – 3 = x + 10.$ $\text{Subtract } x \text{ from both sides: } 3x – 3 = 10.$ $\text{Add 3 to both sides: } 3x = 13 \rightarrow x = \frac{13}{3}.$
2. Set Up Simple Linear Equations from Given Data
• Common Mistake: Misinterpreting the word problem or data, leading to incorrect formulation of the equation.
• Example: $\text{Incorrect: A number decreased by 7 is equal to twice the number increased by 3, set up as } x – 7 = 2x – 3.$ $\text{Correct: Set up as } x – 7 = 2x + 3.$ $\text{Subtract } x \text{ from both sides: } -7 = x + 3.$ $\text{Subtract 3 from both sides: } -10 = x \rightarrow x = -10.$

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