E2.1 Introduction to algebra

Learning objectives

E2.1.1Know that letters can be used to represent generalised numbers.• Know that letters can be used to represent generalised numbers.E2.1.2Substitute numbers into expressions and formulas.• Substitute numbers into expressions and formulas.

A letter can stand for a number whose value is not fixed yet or may change. This lets one algebraic expression describe every value that fits the same relationship.

Relationship	Algebra	Meaning
five copies of a number $n$	$5n$	$5\times n$
three more than $n$	$n+3$	add 3 to the value of $n$
£48 shared equally among $x$ people	$48/x$	each share when $x\ne0$
a number $p$ decreased by 7	$p-7$	subtract 7 from the value of $p$

If one notebook costs £ $p$ , then five notebooks cost £ $5p$ . The same expression works whether $p=2$ , $p=3.50$ or another allowed price. The letter represents the number; it is not an abbreviation for the object.

Within one expression, every occurrence of the same letter has the same value. Different letters represent quantities independently unless the context states a relationship between them.

$5n$ means $5\times n$ , not $5+n$ and not the two-digit number “5n”. Keep the operation stated by the relationship: “three more” gives $n+3$ , while “three times” gives $3n$ .

Substitution means replacing each letter with its given numerical value while keeping every operation in the original expression or formula.

Use this order: (1) write the original expression; (2) replace every letter with its value, putting negative values in brackets; (3) evaluate powers and brackets first; (4) multiply or divide; (5) add or subtract; (6) check that every occurrence was replaced.

$F=2a^2+3b,\qquad a=-3,\ b=4$

Substitute before calculating: $F=2(-3)^2+3(4)$ . Then $(-3)^2=9$ , so $F=2(9)+12=30$ . The brackets ensure the square applies to the whole negative value.

A formula may contain several letters. For $s=\frac12(u+v)t$ with $u=20$ , $v=30$ and $t=7$ , substitution gives $s=\frac12(20+30)(7)=175$ . Keep the grouping shown by the formula.

Do not change signs or operations while substituting. In particular, $(-3)^2=9$ but $-3^2=-9$ because the exponent applies before the leading minus when there are no brackets.

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—E2.2 Algebraic manipulationRead

E2.1 Introduction to algebra

Learning objectives

Use letters to represent general numbers

Substitute values into expressions and formulas

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