Algebra Revision Notes
Subject: Mathematics | Level: GCSE | Exam Board: WJEC
Algebra is the language of mathematics, using letters to represent unknown values and general rules. Mastering algebra unlocks the ability to solve complex problems across the entire specification, making it a critical foundation for exam success.
Revision Notes & Key Concepts
Revision Podcast Transcript
Welcome to your GCSE Maths Revision Podcast. I'm your tutor, and today we're diving deep into one of the most important topics in the entire specification — Algebra. Whether you're sitting your Foundation or Higher paper, algebra is everywhere. It's worth a significant chunk of marks, and the great news is, once you understand the core ideas, you can unlock so many different question types. So grab a pen, get comfortable, and let's get into it. Algebra is essentially the language of mathematics. Instead of working with specific numbers, we use letters — usually x, y, or n — to represent unknown values or general rules. This might sound abstract, but think about it this way: if I told you that three bags of sweets all contain the same number of sweets, and together they contain 24 sweets, you'd probably work out that each bag has 8 sweets. That's algebra. You've just solved 3x equals 24, so x equals 8. Simple as that. Now let's build up through the key concepts you need to master. The first big area is simplifying and manipulating expressions. An expression is a collection of terms — things like 3x, 2y squared, or minus 5. When you simplify, you collect like terms. Like terms are terms with the same letter and the same power. So 3x and 5x are like terms — they both have x to the power 1 — and they add to give 8x. But 3x and 3x squared are NOT like terms, because the powers are different. Examiners will test this distinction, so be careful. Expanding brackets is the next key skill. When you see something like 3 brackets open x plus 4 brackets close, you multiply the 3 by everything inside: 3 times x gives 3x, and 3 times 4 gives 12. So the answer is 3x plus 12. For double brackets, like x plus 2 times x plus 3, you use FOIL — First, Outer, Inner, Last. First: x times x gives x squared. Outer: x times 3 gives 3x. Inner: 2 times x gives 2x. Last: 2 times 3 gives 6. Add them all up: x squared plus 5x plus 6. Candidates who forget the Last term — the 6 — lose a mark. Don't be that candidate. Factorising is the reverse of expanding. You're taking an expression and writing it as a product of factors. For simple expressions like 6x plus 9, you find the highest common factor — which is 3 — and write it as 3 times open bracket 2x plus 3 close bracket. Always check your answer by expanding back out. For quadratics — expressions of the form ax squared plus bx plus c — factorising is a crucial skill. Take x squared plus 5x plus 6. You need two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3. So the factorised form is open bracket x plus 2 close bracket times open bracket x plus 3 close bracket. Examiners award one mark for the correct structure and one mark for the correct values. If the coefficient of x squared is not 1 — say it's 2x squared plus 7x plus 3 — you need to use the ac method or inspection. This is Higher tier content, and it's worth practising carefully. Now let's talk about solving equations. An equation has an equals sign, and your job is to find the value of the unknown. The golden rule is: whatever you do to one side, you must do to the other. Think of the equation as a set of scales — keep it balanced at all times. For linear equations like 2x plus 5 equals 13, subtract 5 from both sides to get 2x equals 8, then divide both sides by 2 to get x equals 4. Always show every step — method marks are available even if you make an arithmetic slip. For equations with brackets, expand first, then solve. For equations with x on both sides, collect the x terms on one side first. For example, 5x minus 3 equals 2x plus 9. Subtract 2x from both sides: 3x minus 3 equals 9. Add 3 to both sides: 3x equals 12. Divide by 3: x equals 4. Forming and solving equations is a favourite exam question style. You might be told that a rectangle has a perimeter of 40 centimetres, with sides of length 3x plus 1 and x plus 5. You'd form the equation 2 times open bracket 3x plus 1 plus x plus 5 close bracket equals 40, simplify, and solve. These questions are worth 3 to 5 marks and reward candidates who set up the equation clearly before solving. Simultaneous equations involve two equations with two unknowns. The elimination method is usually the most reliable. If you have 2x plus y equals 7 and x minus y equals 2, you can add the equations to eliminate y: 3x equals 9, so x equals 3. Substitute back to find y equals 1. Always check your answer by substituting both values into both original equations. The substitution method is preferred when one equation is already solved for a variable, or when one of the equations is quadratic — which is Higher tier content. Sequences and the nth term are another major algebra topic. An arithmetic sequence has a constant difference between terms. The nth term formula is: nth term equals a plus open bracket n minus 1 close bracket times d, where a is the first term and d is the common difference. For the sequence 3, 7, 11, 15, the first term is 3 and the common difference is 4. So the nth term is 3 plus 4n minus 4, which simplifies to 4n minus 1. Check: when n equals 1, you get 3. When n equals 2, you get 7. Correct. For quadratic sequences — Higher tier — the second differences are constant. You'd find the nth term by working with the second difference to identify the coefficient of n squared, then adjusting. Inequalities represent a range of values. The symbols are: less than, greater than, less than or equal to, and greater than or equal to. Solving inequalities is just like solving equations, with one crucial exception: if you multiply or divide by a negative number, you must flip the inequality sign. So if minus 2x is greater than 6, dividing by minus 2 gives x is less than minus 3. This is one of the most common errors in GCSE algebra. On a number line, an open circle means the value is NOT included — used for strict inequalities — and a closed circle means the value IS included — used for less than or equal to and greater than or equal to. Now, let me give you the exam tips and common mistakes section — this is where marks are won and lost. Tip number one: always show your working. In algebra questions, method marks are awarded for correct algebraic steps even if your final answer is wrong. A candidate who writes a correct equation and makes a small arithmetic error can still earn 2 out of 3 marks. A candidate who writes only the wrong final answer earns zero. Tip number two: check your answer. After solving an equation, substitute your answer back into the original equation to verify it works. This takes 10 seconds and can save you from losing marks. Tip number three: read the command word carefully. "Solve" means find the value of x. "Factorise" means write as a product of brackets. "Expand and simplify" means remove the brackets AND collect like terms. "Show that" means you must demonstrate the result algebraically — you cannot just state it. Tip number four: be careful with negatives. Expanding minus 2 times open bracket x minus 3 close bracket gives minus 2x PLUS 6, not minus 6. The negative times negative gives positive. This is the single most common error in algebra. Tip number five: for quadratic equations, if factorising doesn't work neatly, try the quadratic formula. x equals minus b plus or minus the square root of b squared minus 4ac, all divided by 2a. This is given on the formula sheet in most exam boards, but you must know when and how to use it. Tip number six: in simultaneous equations, always label your equations as equation 1 and equation 2. Examiners follow your working more easily, and you're less likely to make errors. Now for our quick-fire recall quiz! I'll ask a question, then give you a moment to think before I give the answer. Ready? Question 1: What does FOIL stand for when expanding double brackets? Think about it... First, Outer, Inner, Last. Question 2: What is the nth term of the sequence 5, 8, 11, 14? Think... The common difference is 3 and the first term is 5, so the nth term is 3n plus 2. Question 3: When solving an inequality, what must you do if you divide by a negative number? Think... You must flip the inequality sign. Question 4: What are the solutions to x squared plus 5x plus 6 equals 0? Think... Factorise to get x plus 2 times x plus 3 equals 0, so x equals minus 2 or x equals minus 3. Question 5: What is the highest common factor of 12x squared and 8x? Think... The HCF is 4x. Brilliant work on those! Let's wrap up with a quick summary of the key things to remember. Number one: collect like terms carefully — only combine terms with the same letter AND the same power. Number two: when expanding double brackets, use FOIL and don't forget the last term. Number three: factorising is the reverse of expanding — always check by expanding back. Number four: solve equations by keeping both sides balanced — show every step. Number five: for simultaneous equations, use elimination by making coefficients match, then add or subtract. Number six: the nth term of an arithmetic sequence is a plus n minus 1 times d — simplify fully. Number seven: when solving inequalities, flip the sign if you multiply or divide by a negative. And the golden rule: ALWAYS show your working. Method marks are your safety net. That's it for today's episode. Algebra is a topic that rewards practice — the more equations you solve, the more automatic these skills become. Work through past paper questions, check your answers against the mark scheme, and pay attention to the examiner's comments. You've got this. Good luck with your revision, and I'll see you in the next episode!
Key Terms & Definitions
- Term
- A single number, a single variable, or numbers and variables multiplied together (e.g., $4$, $x$, $3xy^2$).
- Expression
- A collection of terms separated by plus or minus signs, with NO equals sign (e.g., $2x + 5$).
- Equation
- A mathematical statement showing that two expressions are equal, containing an equals sign (e.g., $2x + 5 = 11$).
- Formula
- A rule written using symbols that describes a relationship between different quantities (e.g., $A = \pi r^2$).
- Identity
- An equation that is true for ALL values of the variables involved, often denoted by a triple equals sign $\equiv$ (e.g., $2(x+3) \equiv 2x + 6$).
- Coefficient
- The number placed in front of a variable, which multiplies it (e.g., in $7x$, 7 is the coefficient).
Worked Examples
Worked Example
Question: Solve the simultaneous equations: $3x + 2y = 18$ $2x - y = 5$
Solution: Step 1: Label the equations (1) and (2). (1) $3x + 2y = 18$ (2) $2x - y = 5$ Step 2: Multiply equation (2) by 2 to make the coefficients of $y$ equal (but opposite signs). (2) $\times 2 \rightarrow 4x - 2y = 10$ (Let's call this equation 3) Step 3: Add equation (1) and equation (3) to eliminate $y$. $(3x + 2y) + (4x - 2y) = 18 + 10$ $7x = 28$ Step 4: Solve for $x$. $x = 4$ Step 5: Substitute $x = 4$ back into equation (2) to find $y$. $2(4) - y = 5$ $8 - y = 5$ $y = 3$ Final answer: $x = 4, y = 3$
Worked Example
Question: The diagram shows a rectangle. The length is $(2x + 5)$ cm and the width is $(x - 1)$ cm. The perimeter of the rectangle is 38 cm. Calculate the value of $x$.
Solution: Step 1: Form an expression for the perimeter. Perimeter = $2 \times \text{length} + 2 \times \text{width}$ Perimeter = $2(2x + 5) + 2(x - 1)$ Step 2: Expand the brackets. Perimeter = $4x + 10 + 2x - 2$ Step 3: Simplify the expression. Perimeter = $6x + 8$ Step 4: Form an equation using the given perimeter of 38 cm. $6x + 8 = 38$ Step 5: Solve the equation. $6x = 30$ $x = 5$ Final answer: $x = 5$
Worked Example
Question: Solve $x^2 - 7x + 10 = 0$
Solution: Step 1: The equation is already equal to zero, so we can factorise the quadratic. We need two numbers that multiply to give $+10$ and add to give $-7$. Step 2: List factor pairs of 10: $1 \times 10$ (sum = 11) $-1 \times -10$ (sum = -11) $2 \times 5$ (sum = 7) $-2 \times -5$ (sum = -7) <-- This is the correct pair. Step 3: Write the quadratic in double brackets. $(x - 2)(x - 5) = 0$ Step 4: Solve for $x$ by setting each bracket to zero. $x - 2 = 0 \Rightarrow x = 2$ $x - 5 = 0 \Rightarrow x = 5$ Final answer: $x = 2$ or $x = 5$
Practice Questions
Question: Simplify fully: $3a + 4b - a + 2b$
Answer:
Question: Expand and simplify: $3(x - 2) + 2(4x + 1)$
Answer:
Question: Solve the inequality: $5 - 2x > 13$
Answer:
Question: The $n^{th}$ term of a sequence is $3n - 2$. Find the first three terms of the sequence.
Answer:
Question: Solve algebraically the simultaneous equations: $y = x^2 + 3x - 5$ $y = 2x + 1$
Answer:
