Algebra

Algebra is the language every other area of this course is written in, so fluency here pays off in functions, calculus and probability. An expression such as 3x² − 5xy has no equals sign; an equation states that two expressions are equal; a formula links several variables; and a system of simultaneous equations is a set of equations that must all be true at once.

When you build an expression from words, define every variable first, for example “let n be the number of tickets sold”.

Function notation. Writing y = f(x) names the rule f. To evaluate f(a), replace every x with a, using brackets around negative values. If f(x) = 2x² − 3x then f(−2) = 2(−2)² − 3(−2) = 8 + 6 = 14.

Rearranging formulae. To make a variable the subject, undo operations in reverse order, doing the same thing to both sides. Keep the subject positive and on the left at the end.

Index laws. For a non-zero base: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿbⁿ, a⁰ = 1 and a⁻ⁿ = 1/aⁿ. A rational index links to surds: a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ. So √(x³) = x^(3/2). To solve an index equation, write both sides with the same base and equate the indices.

Algebraic fractions follow the same rules as numerical fractions: factorise first, find a common denominator, combine numerators, then cancel common factors (never individual terms). State any values that make a denominator zero.

Scientific notation writes a number as m × 10ᵏ with 1 ≤ m less than 10. Multiply by combining the m values and adding the powers of ten, then re-adjust m.