The language of algebra
- Letters stand for numbers. A term is a single part, such as 3x. An expression is a group of terms, such as 3x + 2.
- An equation has an equals sign and can be solved: 3x + 2 = 11. A formula links quantities: A = l × w.
- 3x means 3 × x; x² means x × x; xy means x × y.
Simplifying and substituting
- Collect like terms: 3a + 5b − a + 2b = 2a + 7b. Only terms with the same letters (and powers) can be added.
- Multiply: 2x × 4y = 8xy, and a × a × a = a³.
- Index laws work with letters too: x³ × x⁴ = x⁷, and y⁶ ÷ y² = y⁴.
- Substitute by replacing the letters with numbers: if x = 3 and y = −2, then 2x + 5y = 6 − 10 = −4.
Brackets
- Expand by multiplying every term inside the bracket: 3(x + 4) = 3x + 12, and x(x − 5) = x² − 5x.
- Expand double brackets by multiplying every pair of terms: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6.
- Factorise by taking out the highest common factor: 6x + 9 = 3(2x + 3), and x² + 4x = x(x + 4).
Key terms
- Term
- A single number, letter, or number and letters multiplied together, such as 5x.
- Expression
- A group of terms with no equals sign, such as 2x + 3.
- Equation
- A statement with an equals sign that can be solved, such as 2x + 3 = 9.
- Formula
- A rule linking quantities, such as A = l × w.
- Like terms
- Terms with exactly the same letters and powers.
- Coefficient
- The number in front of a letter, such as the 3 in 3x.
- Substitute
- Replace letters with numbers.
- Expand
- Multiply out brackets.
- Factorise
- Put an expression into brackets by taking out a common factor.