C2.1, C2.5; E2.1, E2.5 - Letters and substitution | Cambridge IGCSE International Mathematics

A machine triples any number you put in, then adds 4. How can you describe every possible output in one line? C2.1 and E2.1 introduce letters and substitution; C2.5 and E2.5 use them to build expressions, equations and formulas from words.

Letters as numbers

Put 2 into the opening machine and its output is $3\times2+4=10$ . Put 5 in and its output is $3\times5+4=19$ . Instead of writing a new calculation for every input, call the input $n$ : the output is $3n+4$ .

The letter represents a number. It is not a label meaning “number” or the name of an object. In one calculation, repeated copies of the same letter have the same value.

Algebraic expression

An algebraic expression combines numbers, letters and operations without an equals sign. For example, $3n+4$ is an expression. It has a numerical value once a value of $n$ is chosen.

Algebra usually leaves out the multiplication sign between a number and a letter: $3n$ means $3\times n$ . Similarly, $ab$ means $a\times b$ , $x^2$ means $x\times x$ , and $\frac{x}{5}$ means $x\div5$ . However, $3+n$ means addition and cannot be shortened to $3n$ .

An equals sign makes a statement about two expressions.

Equation and formula

An equation states that two expressions are equal, for example $3n+4=19$ . A formula is an equation giving a relationship between quantities, for example $P=4s$ , relating the perimeter $P$ of a square to its side length $s$ .

A formula is a kind of equation. The distinction is its use: $P=4s$ describes many squares, while $3n+4=19$ asks which input makes a particular output. The expression $3n+4$ alone does not ask you to find $n$ .

Writing	Meaning
$n+2$	An expression: two more than a number
$n+2=9$	An equation: that expression is equal to 9
$C=2p$	A formula: total cost $C$ of two items costing $p$ each

The power of a letter is that one expression can cover every allowed input. If the situation counts people, the allowed values are whole numbers; if it measures a temperature, negative values may be appropriate.

Key idea

Read the operations, not just the letters. $5x$ , $x+5$ and $x^5$ describe three different calculations.

Decide what the notation actually says.

Writing expressions

Translate the order of a calculation into operations. Start by choosing a letter and stating what it represents. If a whole sum is multiplied or divided, brackets or a fraction bar keep that sum together.

Words	Expression
6 less than $x$	$x-6$
6 divided by $x$	$\frac6x$ , with $x\ne0$
Twice the sum of $x$ and 6	$2(x+6)$
Half the difference between $x$ and 6	$\frac{x-6}{2}$

Order matters for subtraction and division: “6 less than $x$ ” takes 6 away from $x$ , not the other way round.

Worked example 1: expressing a changing cost

Question. A taxi fare has a fixed charge of $3 and a charge of$ 2 for each kilometre. Write an expression for the fare for $d$ kilometres, then a formula using $F$ for the fare in dollars.

Plan. Find the distance charge first, then add the fixed charge once.

\begin{aligned} \text{distance charge}&=2d &&\text{2 dollars for each kilometre}\\ \text{total fare}&=2d+3 &&\text{add the fixed charge} \end{aligned}

Answer. The expression is $2d+3$ ; the formula is $F=2d+3$ .

Check. A 4 km journey costs $4\times2+3=11$ dollars. The expression gives $2\times4+3=11$ . At $d=0$ , it still includes the fixed charge of $3.

Now the total fare is known. This adds an equals sign and turns the cost expression into an equation for the distance.

Worked example 2: expressing a known total

Question. For the same tariff, a journey costs $17. Write an equation for its distance$ d$ kilometres.

Plan. Set the fare expression equal to the given total.

\begin{aligned} \text{fare expression}&=2d+3\\ 2d+3&=17 &&\text{the total is 17 dollars} \end{aligned}

Answer. $2d+3=17$ .

Check. The $3 fixed charge is included once and the remaining charge depends on distance. For example, a 7 km journey gives$ 2\times7+3=17$, consistent with the equation.

Constructing an equation and solving it are different tasks. Here the equation is the required answer; later lessons develop systematic ways to solve it.

Watch out

“Three times the sum of $x$ and 4” is $3(x+4)$ . Writing $3x+4$ adds 4 after multiplying, so it describes a different calculation.

Write a complete equality for the next known total.

Substitution

Substitution replaces each letter with its stated number. Use brackets around a negative replacement, then follow the normal order of operations: powers before multiplication, then addition and subtraction.

Substitution

To substitute a value into an expression or formula is to replace every occurrence of its letter with that value and evaluate the resulting calculation.

For example, if $x=-2$ , then $x^2=(-2)^2=4$ , while $-x^2=-(-2)^2=-4$ . The minus sign before the square is an extra operation; it is not part of the squared letter.

Worked example 3: a positive input

Question. Find the value of $2x^2-3x+1$ when $x=3$ .

Plan. Replace both copies of $x$ , evaluate the square and products, then combine.

\begin{aligned} 2x^2-3x+1&=2(3)^2-3(3)+1 &&\text{replace each }x\\ &=2(9)-9+1 &&\text{square first}\\ &=18-9+1\\ &=10 \end{aligned}

Answer. $10$ .

Check. Reverse the last operations: $10-1+3\times3=18$ , which equals $2\times3^2$ . This recovers the squared contribution from the proposed answer.

Only the sign of the input changes next. Brackets keep a negative number together and reveal the subtraction of a negative product.

Worked example 4: a negative input

Question. Find the value of $2x^2-3x+1$ when $x=-3$ .

Plan. Substitute $(-3)$ everywhere, then handle the square and signs separately.

\begin{aligned} 2x^2-3x+1&=2(-3)^2-3(-3)+1\\ &=2(9)-(-9)+1 &&\text{square positive, product negative}\\ &=18+9+1 &&\text{subtract a negative}\\ &=28 \end{aligned}

Answer. $28$ .

Check. Changing 3 to $-3$ leaves $2x^2$ unchanged. The term $-3x$ changes from $-9$ to $+9$ , increasing the previous result by 18: $10+18=28$ .

The same replacement works in a formula with several letters. If $T=\frac{a+b}{c}$ , insert all the given values in their places and calculate the whole numerator before dividing. The formula is undefined when $c=0$ : division by zero has no value.

Watch out

With $x=-3$ , writing $2\times-3^2$ loses the intended brackets. Write $2(-3)^2=18$ . Also replace every occurrence of the letter, not just the first.

Use the signed values in this formula.

Quick Recap

Letters represent numbers; the same letter has the same value within a calculation.
An expression has no equals sign. An equation states equality; a formula relates quantities.
Translate the order of operations carefully, using brackets for a whole sum.
Substitute into every occurrence and bracket negative values before calculating.

The opening machine has output $3n+4$ , or the formula $y=3n+4$ if $y$ names the output.

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C2.1, C2.5; E2.1, E2.5 - Letters and substitution

Letters as numbers

Writing expressions

Substitution

Quick Recap

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