How do I solve linear equations with unknowns on both sides?

To solve equations like 3x + 2 = 2x + 5, start by getting all the x terms on one side and the constants on the other. Subtract 2x from both sides to get x + 2 = 5, then subtract 2 to get x = 3. Always check by substituting back into the original equation: 3(3)+2 = 11 and 2(3)+5 = 11, so it works.

What is the difference between an expression, an equation, and an identity?

An expression is a collection of terms without an equals sign, like 3x + 2. An equation has an equals sign and can be solved, like 3x + 2 = 11. An identity is true for all values of the variable, often shown with a triple equals sign (≡), like 3(x + 2) ≡ 3x + 6. In an identity, both sides are equivalent expressions.

How do I factorise a quadratic when the coefficient of x^2 is not 1?

For quadratics like 2x^2 + 7x + 3, use the 'ac method'. Multiply the coefficient of x^2 (2) by the constant (3) to get 6. Find two numbers that multiply to 6 and add to 7 (1 and 6). Rewrite the middle term: 2x^2 + 1x + 6x + 3. Factorise in pairs: x(2x+1) + 3(2x+1) = (2x+1)(x+3). Always expand to check.

What is the quadratic formula and when should I use it?

The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a) for equations of the form ax² + bx + c = 0. Use it when factorising is difficult or impossible, especially when the discriminant (b² - 4ac) is not a perfect square. For example, solve 2x² - 4x - 3 = 0: a=2, b=-4, c=-3. Discriminant = (-4)² - 4(2)(-3) = 16 + 24 = 40. So x = [4 ± √40] / 4 = [4 ± 2√10] / 4 = 1 ± (√10)/2.

How do I solve quadratic inequalities like x² - 5x + 6 < 0?

First, solve the corresponding equation x² - 5x + 6 = 0 to find critical values. Factorise: (x-2)(x-3)=0, so x=2 or x=3. Sketch the quadratic graph (it opens upwards). The inequality <0 means the part of the graph below the x-axis, which is between the roots. So the solution is 2 < x < 3. For ≤0, include the endpoints: 2 ≤ x ≤ 3. Always check a test point, e.g., x=2.5 gives (0.5)(-0.5) = -0.25 < 0, correct.

What is the nth term of a linear sequence and how do I find it?

The nth term of a linear sequence is an expression that gives the value of the term at position n. For example, the sequence 5, 8, 11, 14,... has a common difference of 3. The nth term is 3n + 2 because when n=1, 3(1)+2=5; n=2 gives 8, etc. To find it, calculate the difference (d), then the term is dn + (first term - d). So here, d=3, first term=5, so nth term = 3n + (5-3) = 3n+2.

Algebra

Q: How do I solve quadratic inequalities like x² - 5x + 6 < 0?

First, solve the corresponding equation x² - 5x + 6 = 0 to find critical values. Factorise: (x-2)(x-3)=0, so x=2 or x=3. Sketch the quadratic graph (it opens upwards). The inequality <0 means the part of the graph below the x-axis, which is between the roots. So the solution is 2 < x < 3. For ≤0, include the endpoints: 2 ≤ x ≤ 3. Always check a test point, e.g., x=2.5 gives (0.5)(-0.5) = -0.25 < 0, correct.

Q: What is the nth term of a linear sequence and how do I find it?

The nth term of a linear sequence is an expression that gives the value of the term at position n. For example, the sequence 5, 8, 11, 14,... has a common difference of 3. The nth term is 3n + 2 because when n=1, 3(1)+2=5; n=2 gives 8, etc. To find it, calculate the difference (d), then the term is dn + (first term - d). So here, d=3, first term=5, so nth term = 3n + (5-3) = 3n+2.

AQA

GCSE

Algebra involves the use of symbols and notation to represent mathematical relationships, expressions, and functions. Students learn to manipulate algebraic expressions, solve various types of equations and inequalities, and interpret graphical representations of linear, quadratic, and other functions.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

Topic Overview

Algebra is a fundamental branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and expressions. In the AQA GCSE Mathematics course, algebra is a major component, covering topics from simplifying expressions and solving linear equations to working with quadratics, inequalities, and sequences. Mastering algebra is essential because it provides the tools to model real-world situations, solve problems systematically, and forms the basis for more advanced topics like functions and graphs.

Why does algebra matter? Beyond the exam, algebraic thinking develops logical reasoning and problem-solving skills that are valuable in everyday life and many careers, including science, engineering, economics, and technology. In the AQA GCSE, algebra appears in both the non-calculator and calculator papers, often in multi-step problems that require you to manipulate expressions, solve equations, and interpret graphs. A strong grasp of algebra can significantly boost your overall grade, as it typically accounts for about 20-30% of the total marks.

Algebra builds on your knowledge of number operations and arithmetic, extending them into abstract reasoning. You will learn to use algebraic notation fluently, rearrange formulas, solve equations of increasing complexity, and understand the relationship between algebraic and graphical representations. This topic is not just about memorising rules; it's about understanding why those rules work so you can apply them flexibly in unfamiliar contexts.

Key Concepts

Core ideas you must understand for this topic

→Simplifying expressions: Collecting like terms, expanding brackets, and factorising (including difference of two squares and quadratics).
→Solving linear equations: Using inverse operations to isolate the variable, including equations with unknowns on both sides and brackets.
→Solving quadratic equations: Factorising, completing the square, and using the quadratic formula. Understanding the discriminant and its role in determining the number of real roots.
→Inequalities: Representing solutions on number lines and solving linear and quadratic inequalities, including compound inequalities.
→Sequences: Finding the nth term of linear and quadratic sequences, and using term-to-term and position-to-term rules.

What You Need to Demonstrate

Key skills and knowledge for this topic

Correct use and interpretation of algebraic notation
Accurate substitution of numerical values into formulae
Correct simplification of expressions by collecting like terms and using laws of indices
Correct expansion of brackets and factorisation of expressions
Accurate solution of linear and quadratic equations
Correct identification of gradients and intercepts from linear graphs
Accurate plotting of functions and interpretation of graphical features
Correct derivation of equations from word problems

Marking Points

Key points examiners look for in your answers

Correct use and interpretation of algebraic notation
Accurate substitution of numerical values into formulae
Correct simplification of expressions by collecting like terms and using laws of indices
Correct expansion of brackets and factorisation of expressions
Accurate solution of linear and quadratic equations
Correct identification of gradients and intercepts from linear graphs
Accurate plotting of functions and interpretation of graphical features
Correct derivation of equations from word problems

Examiner Tips

Expert advice for maximising your marks

💡Always show your working out, as method marks are awarded even if the final answer is incorrect
💡Check your answers by substituting values back into the original equation
💡Ensure you are familiar with the calculator functions for solving equations if permitted
💡Read the question carefully to see if an exact answer (e.g., in terms of pi or surds) is required
💡Use a ruler for drawing straight-line graphs and ensure axes are clearly labelled
💡Show all your working: Even if you make a mistake, you can still get method marks. For example, in solving equations, write each step clearly. If your final answer is wrong but your method is correct, you may still earn up to 3 out of 4 marks.
💡Check your answers: Substitute your solution back into the original equation to verify it works. For quadratics, you can also use the discriminant to check if your solutions are real. This simple habit can catch many errors.
💡Read the question carefully: Look for keywords like 'simplify', 'solve', 'factorise', or 'hence'. 'Hence' means you must use your previous answer. Also, pay attention to whether the question asks for exact values or decimals, and whether you need to show inequalities on a number line.

Common Mistakes

Pitfalls to avoid in your exam answers

Errors in sign when expanding brackets or solving equations
Confusing the rules for indices (e.g., adding instead of multiplying)
Incorrectly identifying the gradient or intercept from a linear equation
Failing to include all solutions for quadratic equations
Misinterpreting inequality signs on number lines or graphs
Errors in substitution, particularly with negative numbers
Misconception: When expanding brackets like 3(x + 4), students often write 3x + 4, forgetting to multiply the constant term. Correction: Remember to multiply every term inside the bracket by the term outside: 3(x + 4) = 3x + 12.
Misconception: Solving equations like 2x + 3 = 11, students sometimes subtract 3 from the left but add 3 to the right. Correction: Whatever you do to one side, you must do to the other. Subtract 3 from both sides to get 2x = 8, then divide by 2.
Misconception: When factorising quadratics like x^2 + 5x + 6, students may write (x + 2)(x + 3) but then expand incorrectly, or they forget to check the signs. Correction: Always expand your factorisation to verify it matches the original expression.

Frequently Asked Questions

Common questions students ask about this topic

Before You Start

Prior knowledge that will help with this topic

•Basic number operations: addition, subtraction, multiplication, division, and order of operations (BIDMAS/BODMAS).
•Understanding of negative numbers and fractions, as they frequently appear in algebraic manipulation.
•Familiarity with basic geometric concepts like area and perimeter, as algebra is often applied in word problems involving shapes.

Study Guide Available

Comprehensive revision notes & examples

Read Study Guide

Likely Command Words

How questions on this topic are typically asked

Solve

Simplify

Expand

Factorise

Plot

Sketch

Rearrange

Show that

Ready to test yourself?

Practice questions tailored to this topic

Algebra

Topic Overview

Key Concepts

What You Need to Demonstrate

Marking Points

Examiner Tips

Common Mistakes

Frequently Asked Questions

Before You Start

Study Guide Available

Likely Command Words

Ready to test yourself?

Related Topics in AQA GCSE Mathematics

E2E stub concept

Geometry and measures

Number

Probability

Topic Synopsis

Key Concepts & Core Principles

Exam Tips & Revision Strategies

Common Misconceptions & Mistakes to Avoid

Examiner Marking Points

Algebra

Topic Overview

Key Concepts

What You Need to Demonstrate

Marking Points

Examiner Tips

Common Mistakes

Frequently Asked Questions

How do I solve linear equations with unknowns on both sides?

What is the difference between an expression, an equation, and an identity?

How do I factorise a quadratic when the coefficient of x^2 is not 1?

What is the quadratic formula and when should I use it?

How do I solve quadratic inequalities like x² - 5x + 6 < 0?

What is the nth term of a linear sequence and how do I find it?

Before You Start

Study Guide Available

Likely Command Words

Ready to test yourself?

Related Topics in AQA GCSE Mathematics

E2E stub concept

Geometry and measures

Number

Probability