Edexcel GCSE Maths: Solving Linear Equations

Edexcel GCSE Maths: Solving Linear Equations

This tutorial will guide you through solving linear equations, covering one-step, two-step, and multi-step equations. We'll explore how to apply these skills to word problems, algebraic manipulation, and problem-solving in various contexts.

What are Linear Equations?

Linear equations are equations where the highest power of the variable is 1. They are represented by a straight line when graphed. For example:

• 2x + 3 = 7
• y - 5 = 10
• 3a + 2 = 4a - 1

Solving One-Step Equations

To solve one-step equations, we aim to isolate the variable on one side of the equation. This is achieved using inverse operations:

1. Addition and Subtraction:

• Example: x + 5 = 8
  • Subtract 5 from both sides: x + 5 - 5 = 8 - 5
  • Simplify: x = 3

2. Multiplication and Division:

• Example: 2y = 10
  • Divide both sides by 2: 2y / 2 = 10 / 2
  • Simplify: y = 5

Solving Two-Step Equations

Two-step equations involve combining addition/subtraction and multiplication/division operations.

1. Example: 3x - 2 = 10 * Step 1: Add 2 to both sides: 3x - 2 + 2 = 10 + 2 * Step 2: Divide both sides by 3: 3x / 3 = 12 / 3 * Solution: x = 4

Solving Multi-Step Equations

Multi-step equations involve more than two steps. The key is to simplify the equation by combining like terms and isolating the variable.

1. Example: 4(x + 2) - 3 = 5x + 1 * Step 1: Expand the brackets: 4x + 8 - 3 = 5x + 1 * Step 2: Combine like terms: 4x + 5 = 5x + 1 * Step 3: Subtract 4x from both sides: 5 = x + 1 * Step 4: Subtract 1 from both sides: 4 = x * Solution: x = 4

Word Problems

Linear equations are useful in solving real-world problems. To solve a word problem using equations:

1. Identify the unknown variable: Assign a letter (e.g., x, y) to represent the unknown quantity.
2. Translate the words into an equation: Use the given information to write an equation.
3. Solve the equation: Apply the techniques discussed earlier.
4. Interpret the solution: State the answer in the context of the problem.

Example: "Sarah bought 3 apples and 2 oranges for £5. If an apple costs £1, how much does an orange cost?"

• Unknown variable: Let y represent the cost of an orange.
• Equation: 3(£1) + 2y = £5
• Solve:
  • 3 + 2y = 5
  • 2y = 2
  • y = 1
• Answer: An orange costs £1.

Algebraic Manipulation

When solving complex equations, you may need to manipulate the equation to simplify it:

1. Expanding brackets: Use the distributive property to multiply the term outside the bracket by each term inside.
2. Combining like terms: Combine terms with the same variable and exponent.
3. Factoring: Express the equation in terms of common factors.
4. Using inverse operations: Apply inverse operations to isolate the variable.

Problem-Solving

Linear equations are fundamental for solving a variety of problems across different fields. They are used in:

• Finance: Calculating interest, loans, and investments.
• Science: Modeling physical phenomena, such as motion and temperature.
• Engineering: Designing structures and systems.
• Computer science: Programming and data analysis.

Practice Makes Perfect

Solving linear equations requires practice. Use textbooks, online resources, and practice questions to solidify your understanding. Start with simple examples and gradually move towards more complex problems. Remember to review your work and identify areas where you need improvement.