Equations

2026 What You Need to Learn (Syllabus Objectives)

Core Content:

  1. Construct simple expressions, equations and formulas
  2. Solve linear equations in one unknown
  3. Solve simultaneous linear equations in two unknowns
  4. Change the subject of simple formulas

Extended Content (you also need to know all Core content): 5. Construct expressions, equations and formulas (more complex situations) 6. Solve fractional equations with numerical and linear algebraic denominators 7. Solve simultaneous equations involving one linear and one non-linear equation 8. Solve quadratic equations by factorisation, completing the square, and using the quadratic formula 9. Change the subject of formulas (including cases where the subject appears twice or involves powers and roots)

Part A: Core Content

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1. Constructing Simple Expressions from Words

An expression is a mathematical phrase that contains numbers, letters (called variables), and operation symbols like +, −, ×, and ÷. It does not have an equals sign.

Common Phrases and Their Expressions

Let's use the letter x to represent an unknown number. Here's how to translate everyday phrases into algebra:

PhraseExpressionExplanation
2 more than xx + 2Add 2 to x
3 less than xx − 3Subtract 3 from x
Double x2xMultiply x by 2
Half of xx/2 or ½xDivide x by 2
5 times x5xMultiply x by 5
The sum of x and 7x + 7Add x and 7 together
The product of x and 44xMultiply x and 4

Key Words for Operations

  • Addition: sum, total, more than, increase, plus
  • Subtraction: difference, less than, decrease, minus, fewer than
  • Multiplication: product, times, double, triple, lots of
  • Division: shared, split, halved, quartered, divided by

Using Brackets

Brackets help keep the order of operations correct.

Example 1: "Add 1 to x, then multiply the result by 3"

  • This means: (x + 1) × 3, which we write as 3(x + 1)

Example 2: "Multiply x by 3, then add 1"

  • This means: 3x + 1

Notice how these are different! Brackets show what operation happens first.

Example 3: Write an expression for a number that is 2 more than n.

  • Answer: n + 2

Example 4: Write an expression for the product of two consecutive even numbers.

  • Let the first even number be x
  • The next consecutive even number is x + 2
  • The product means we multiply them together
  • Answer: x(x + 2) or x² + 2x

Choosing Your Variable Wisely

Sometimes you need to represent two or more related quantities.

Example: Adam is 10 years younger than Barry.

  • Option 1: Let Barry's age be x. Then Adam's age is x − 10.
  • Option 2: Let Adam's age be x. Then Barry's age is x + 10.

Both work! But sometimes one choice makes the algebra easier.

Example: Adam's age is half of Barry's age.

  • Better choice: Let Adam's age be x. Then Barry's age is 2x.
  • Harder choice: Let Barry's age be x. Then Adam's age is x/2.

The first choice avoids fractions, which usually makes calculations simpler.

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