C2.1 Introduction to algebra

Learning objectives

C2.1.1Know that letters can be used to represent generalised numbers.• Know that letters can be used to represent generalised numbers.C2.1.2Substitute numbers into expressions and formulas.• Substitute numbers into expressions and formulas.

A letter can represent any number allowed by a situation. An algebraic expression then describes one calculation that works for every permitted value of that letter.

Notation	Meaning	Example in context
$x$	one variable quantity	number of pens
$60x$	60 multiplied by $x$	cost in cents of $x$ pens at 60 cents each
$60x+29y$	sum of two variable costs	cost of $x$ pens and $y$ rulers
$d/n$	one total shared among $n$ equal parts	cost of one bag when $n$ bags cost $d$ dollars

Identify what each letter counts or measures, attach its rate or multiplier, then combine parts using the action in the context. If a team earns 3 points per win and 1 per draw, $w$ wins and $d$ draws give $3w+d$ points.

Order matters when quantities enter or leave. If a train starts with $p$ passengers, then $x$ get off and $y$ get on, the new number is $p-x+y$ . The expression remains general because it works for any valid $p$ , $x$ and $y$ .

A formula names the quantity produced by an expression. A car hire charge of 56 dollars per day plus a fixed 436 dollars can be written $C=56d+436$ , where $C$ is the total cost for $d$ days.

Adjacent symbols indicate multiplication: $60x$ means $60\times x$ , not the two-digit number '60x'. A letter does not always mean one fixed unknown; it may vary across cases. Keep units consistent before forming a general expression.

Substitution replaces each letter with its given numerical value while preserving the original operations, brackets and powers.

Step	Action	Check
1	write the expression or formula unchanged	every symbol is present
2	replace each letter with its value in brackets	negative values stay grouped
3	evaluate powers, then multiplication/division, then addition/subtraction	follow operation order
4	attach the required unit and apply requested rounding	interpret the result

For $s=ut+\dfrac12at^2$ with $u=5.2$ , $t=7$ and $a=1.6$ , substitute first: $s=5.2(7)+\dfrac12(1.6)(7)^2=36.4+39.2=75.6$ . The square applies to the substituted value of $t$ .

Brackets protect signs. For $8x-3y$ with $x=5$ and $y=-2$ , write $8(5)-3(-2)=40+6=46$ . Without brackets, the second negative can be lost.

If a formula needs an intermediate quantity, calculate it explicitly. For $M=2.5AT$ , a rectangle 1.9 m by 0.6 m has $A=1.9\times0.6=1.14$ m²; with $T=8$ , $M=2.5(1.14)(8)=22.8$ kg.

Substitution evaluates a given expression; it does not change the formula or solve for a different letter. In $3t^2$ , square only $t$ before multiplying by 3, and for $t=-4$ use $3(-4)^2$ , not $3(-4^2)$ .

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—C2.2 Algebraic manipulationRead

C2.1 Introduction to algebra

Learning objectives

Use letters to describe general quantities

Substitute values into expressions and formulas

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