How to Revise Algebra
A focused plan for revising algebra: the core skills, common errors, and how to practise.
Algebra uses letters to represent numbers so that you can describe relationships and solve problems in general. Most algebra errors come from a few repeated slips, not from the big ideas. Revising well means practising the core skills and checking for those slips.
Key concepts
A variable is a symbol, usually a letter, standing for a number. An expression combines numbers and variables with operations, such as 3x + 5. An equation states that two expressions are equal, such as 3x + 5 = 20. To solve an equation is to find the value of the variable that makes it true.
A coefficient is the number multiplying a variable. In 3x + 5, the coefficient of x is 3.
The skills to revise, in order
- Simplifying expressions. Collect like terms: 4x + 3x = 7x, but 4x + 3 cannot be simplified further.
- Expanding brackets. a(b + c) = ab + ac. Multiply every term inside the bracket. For example, 3(x − 4) = 3x − 12.
- Factorising. The reverse of expanding. 6x + 9 = 3(2x + 3).
- Solving linear equations. See solving linear equations step by step.
- Solving quadratic equations. See solving quadratic equations.
- Rearranging formulas. Treat the target letter as the unknown and undo operations in reverse order.
- Inequalities. Solve like equations, but reverse the inequality sign when multiplying or dividing by a negative number.
Worked example: rearranging a formula
Make r the subject of C = 2πr.
- Divide both sides by 2π.
- r = C ÷ (2π).
Worked example: an inequality
Solve −2x + 3 < 11.
- Subtract 3: −2x < 8.
- Divide by −2 and reverse the sign: x > −4.
Check with x = 0, which should work: −2(0) + 3 = 3, and 3 < 11. True.
Common mistakes
- Sign errors. −(x − 3) = −x + 3, not −x − 3.
- Doing something to only one side of an equation.
- Forgetting to multiply every term when expanding.
- Treating (a + b)² as a² + b². In fact (a + b)² = a² + 2ab + b².
- Not checking the answer by substituting it back.
How to practise
Do a small set of questions on one skill, check each answer, and write down which kind of error you made. Move on only when you can do a set of five without errors. Use the mathematics practice questions.
Quick recap
Know the vocabulary, simplify and expand carefully, solve equations by doing the same thing to both sides, reverse the sign for negative multipliers in inequalities, and check by substitution.