Maths · 2 Algebra

2.1 Introduction to algebra

Before you start

The order of operations: brackets first, then powers, then $\times$ and $\div$ , then $+$ and $-$ .
Arithmetic with negative numbers, e.g. $4 - (- 3) = 7$ and $(- 2) \times (- 5) = 10$ .
Squares and cubes of small whole numbers, e.g. $4^{2} = 16$ and $2^{3} = 8$ .
Multiplying and dividing decimals and simple fractions without a calculator, e.g. $4 \times 1.35 = 5.4$ and $4 \times \frac{1}{2} = 2$ .

Syllabus objectives covered (2)

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

Part 1 of 6

What a letter stands for

In algebra a letter stands for a number. It never stands for an object. If $p$ is the price of a pen in dollars, then $p$ is a number such as $1.35$ . It is not 'a pen'.
Sometimes the letter stands for one number that you do not know yet. In $3 x + 1 = 10$ only one value works, $x = 3$ . Finding it is called solving, and you meet it in a later subtopic.
Sometimes the letter stands for any number. This is a generalised number. The statement below is true whatever numbers you put in for $a$ and $b$ , so it describes every pair of numbers at once.
$a + b = b + a$
A formula uses letters in the same way. $P = 2 l + 2 w$ gives the perimeter of every rectangle, because $l$ and $w$ can be any length and any width. Put in the measurements of one rectangle, for example $l = 6$ and $w = 4$ , and you get the perimeter of that rectangle: $P = 2 \times 6 + 2 \times 4 = 20$ .
You can test a statement about generalised numbers by putting in numbers. Try $a - b = b - a$ with $a = 5$ and $b = 2$ . The left side is $5 - 2 = 3$ . The right side is $2 - 5 = - 3$ . One pair of numbers that fails is enough to show that the statement is not true in general.
The opposite does not work. A few numbers that pass do not prove a statement, because there are always more numbers that you did not try. To show that something is true for every value, you use the letters themselves. The first worked example below shows how.

Part 2 of 6

How algebra is written

Algebra leaves out the multiplication sign. $3 n$ means $3 \times n$ , and $a b$ means $a \times b$ . Write the number first: $3 n$ , not $n 3$ . Write $n$ , not $1 n$ .
A power shows that the letter is multiplied by itself: $n \times n = n^{2}$ and $n \times n \times n = n^{3}$ .
The trap is to mix up $n^{2}$ and $2 n$ . They are different numbers. With $n = 5$ , $n^{2} = 5 \times 5 = 25$ , but $2 n = 2 \times 5 = 10$ .
$n^{2} = n \times n 2 n = n + n$
In the same way, $3 n$ and $n + 3$ are different. With $n = 4$ , $3 n = 12$ and $n + 3 = 7$ . Read $3 n$ as 'three lots of $n$ ' and $n + 3$ as ' $n$ , then three more'.
Division is written as a fraction: $a \div b$ is $\frac{a}{b}$ . The fraction line also acts as a bracket. $\frac{a + b}{2}$ means the whole of $a + b$ , divided by $2$ .
An expression is letters and numbers joined by operations, with no equals sign, such as $5 x - 2 y + 7$ . Its terms are $5 x$ , $- 2 y$ and $7$ . The sign in front of a term belongs to that term, so the coefficient of $y$ is $- 2$ , not $2$ .
A formula has an equals sign and links quantities, such as $A = l w$ for the area of a rectangle. The letter on its own, here $A$ , is the subject of the formula.

Part 3 of 6

From words to an expression

Before you write any algebra, say what each letter stands for, with its unit: 'let $x$ be the cost of one adult ticket, in dollars'. A letter with no stated meaning gives an expression that nobody can check, including you.
Then change the words into algebra one phrase at a time. 'The sum of $a$ and $b$ ' is $a + b$ . 'The product of $a$ and $b$ ' is $a b$ . 'Half of $x$ ' is $\frac{x}{2}$ .
Take care with the order in a subtraction. ' $5$ less than $y$ ' and 'subtract $5$ from $y$ ' both mean $y - 5$ : start with $y$ , then take $5$ away. $5 - y$ is a different number. With $y = 8$ , $y - 5 = 3$ but $5 - y = - 3$ .
Use brackets when an operation acts on a whole group. 'Add $a$ and $b$ , then multiply by $3$ ' is $3 (a + b)$ . If you write $3 a + b$ , only $a$ is multiplied by $3$ .
Letters can also describe a type of number. If $n$ is any whole number, then $n + 1$ is the next whole number. $2 n$ is always even, because it is $2$ times a whole number. $2 n + 1$ is always odd, because it is one more than an even number.
An expression contains no equals sign and no words. Put the unit in your sentence, outside the algebra: 'the total cost is $3 x + 5 y$ dollars'.

Part 4 of 6

Substituting into an expression

To substitute means to replace each letter with the number that it stands for, and then work out the value.
Put back the hidden multiplication sign first. With $a = 4$ , $3 a$ is $3 \times 4 = 12$ . It is not $34$ : writing the digits side by side makes a new number, and nothing has been multiplied.
Write the line with the numbers in place before you calculate. For $2 + 3 a^{2}$ with $a = 4$ , write $2 + 3 \times 4^{2}$ . This line shows your method. If you make a slip in the arithmetic after it, the method is still on the page.
Then use the order of operations: the power first, then the multiplication, then the addition.
$2 + 3 \times 4^{2} = 2 + 3 \times 16 = 2 + 48 = 50$
A power applies only to the thing it is attached to. In $3 a^{2}$ only $a$ is squared, so $3 a^{2} = 3 \times 16 = 48$ . The expression ${(3 a)}^{2}$ is different: it is $12^{2} = 144$ .
With more than one letter, substitute all of them in the same line. With $a = 4$ and $b = 7$ :
$a b - 2 b = 4 \times 7 - 2 \times 7 = 28 - 14 = 14$

Part 5 of 6

Negative numbers, powers and fractions

When a value is negative, put it in brackets as you substitute. The brackets keep the minus sign with the number. With $x = - 3$ , $5 x = 5 \times (- 3) = - 15$ .
A negative number squared is positive: ${(- 3)}^{2} = (- 3) \times (- 3) = 9$ . A negative number cubed stays negative: ${(- 3)}^{3} = - 27$ .
Subtracting a negative number is the same as adding. For $x^{2} - 4 x$ with $x = - 3$ :
${(- 3)}^{2} - 4 \times (- 3) = 9 - (- 12) = 9 + 12 = 21$
The trap is the calculator. If you type $- 3^{2}$ , a calculator that follows the order of operations squares the $3$ first and then applies the minus sign. It shows $- 9$ . Type ${(- 3)}^{2}$ , with the brackets, and it shows $9$ .
Many calculators have a negative key, $(-)$ , that is a different key from subtract. If your calculator has one, use it to enter a negative number.
$- x^{2}$ means $- (x^{2})$ : square first, then make the result negative. With $x = - 3$ , $- x^{2} = - (9) = - 9$ . Compare this with $x^{2} = 9$ . The position of the minus sign changes the answer.
A fraction line groups everything above it and everything below it. Work out the top and the bottom separately, then divide. For $\frac{a + b}{2 c}$ with $a = 7$ , $b = 5$ and $c = 3$ , the top is $12$ and the bottom is $6$ , so the value is $2$ .
A calculator cannot see a fraction line that you write on paper. Type $(7 + 5) \div (2 \times 3)$ , with brackets round the whole top and the whole bottom. If you type $7 + 5 \div 2 \times 3$ , it gives $14.5$ , because it divides and multiplies before it adds.

Part 6 of 6

Substituting into a formula

You substitute into a formula in the same way. Each letter stands for a quantity, and the result is the value of the subject. For $v = u + a t$ with $u = 3$ , $a = 2$ and $t = 5$ :
$v = 3 + 2 \times 5 = 13$
Give the unit if the question gives one. If the question says that $v$ is a speed in m/s, the answer is $13$ m/s.
If the question prints a formula, use exactly that formula. Papers 2 and 4 also have a list of formulas on page 2, but not every formula is on it. You must recall the speed, distance and time relationship, percentage calculations, and the area and perimeter formulas (the area of a triangle is the exception).
Sometimes you know the subject and must find a different letter. Substitute every value that you know first. Then solve the equation that is left. For $v = u + a t$ with $v = 31$ , $u = 7$ and $t = 6$ :
$31 = 7 + 6 a$
Subtract $7$ from both sides to get $24 = 6 a$ . Then divide both sides by $6$ , because $6$ is what multiplies $a$ . So $a = 4$ . When you substitute first, you work with numbers, which are easier to handle than letters.
Check by putting the answer back into the formula: $7 + 4 \times 6 = 31$ . This is correct.
If the answer is not exact, give it to 3 significant figures, unless the question asks for something different. Keep the full value on your calculator and round once, at the end. For $A = π r^{2}$ with $r = 2.6$ , $A = π \times {2.6}^{2} = 21.237 \dots$ , so $A = 21.2$ to 3 significant figures.
Use the $π$ key on your calculator, or $3.142$ . If an answer is exact to 4 or 5 significant figures, such as $268.76$ , give it in full and do not round it.

Worked examples

Example 1. $n$ stands for any whole number. (a) Write down an expression for the next whole number after $n$ . (b) Show that the sum of $n$ and the next whole number is always odd.

Show the working

(a) The next whole number is one more than $n$ , so it is $n + 1$ . This works for every value: if $n = 7$ , the next number is $8$ . If $n = 30$ , it is $31$ .
(b) Write the sum with the letters. Use a bracket so that the second number stays together.
$n + (n + 1)$
Two lots of $n$ make $2 n$ , so the sum is $2 n + 1$ .
$n + (n + 1) = 2 n + 1$
Now give the reason. $2 n$ is $2$ times a whole number, so it is even. One more than an even number is odd. So $2 n + 1$ is odd.
You can check with numbers: $7 + 8 = 15$ , and $2 \times 7 + 1 = 15$ . But the check is not the proof. The algebra is the proof, because $n$ stands for every whole number at the same time.

Answer: (a) $n + 1$ (b) $n + (n + 1) = 2 n + 1$ . $2 n$ is even, and one more than an even number is odd, so the sum is odd for every whole number $n$ .

Example 2. A pen costs $p$ dollars and a notebook costs $q$ dollars. Ama buys $4$ pens and $1$ notebook. She pays with a $$ 20$ note. (a) Write an expression for the total cost, in dollars. (b) Write an expression for her change, in dollars. (c) Find her change when $p = 1.35$ and $q = 2.80$ .

Show the working

State what the letters mean: $p$ is the price of one pen in dollars, and $q$ is the price of one notebook in dollars.
(a) $4$ pens cost $4 \times p = 4 p$ dollars. Add the notebook.
$4 p + q$
(b) The change is $20$ minus the whole cost. Put the cost in a bracket, because all of it is taken away.
$20 - (4 p + q) = 20 - 4 p - q$
Notice what happens without the bracket: $20 - 4 p + q$ would add the notebook, not subtract it.
(c) Substitute. $4 p = 4 \times 1.35 = 5.40$ , so the cost is $5.40 + 2.80 = 8.20$ dollars.
$20 - (4 \times 1.35 + 2.80) = 20 - 8.20 = 11.80$
Check that the answer makes sense: the change is less than $$ 20$ and more than $$ 0$ .

Answer: (a) $4 p + q$ (b) $20 - 4 p - q$ (c) $$ 11.80$

Example 3. Do not use a calculator. Given that $a = - 3$ , $b = 4$ and $c = \frac{1}{2}$ , find the value of (a) $5 a - 2 b$ (b) $a b^{2}$ (c) $\frac{b - a}{4 c}$ .

Show the working

(a) Put the negative value in brackets, then multiply before you subtract.
$5 \times (- 3) - 2 \times 4 = - 15 - 8 = - 23$
(b) The power is attached to $b$ only. Square $b$ first, then multiply by $a$ .
$(- 3) \times 4^{2} = (- 3) \times 16 = - 48$
If you squared $a b$ instead, you would get ${(- 12)}^{2} = 144$ . That is the value of ${(a b)}^{2}$ , which is a different expression.
(c) The fraction line groups the top and the bottom. Work out each one separately. Top: $4 - (- 3) = 4 + 3 = 7$ . Bottom: $4 \times \frac{1}{2} = 2$ .
Then divide.
$\frac{7}{2} = 3.5$

Answer: (a) $- 23$ (b) $- 48$ (c) $\frac{7}{2} = 3.5$

Example 4. The formula $C = \frac{5 (F - 32)}{9}$ changes a temperature of $F$ degrees Fahrenheit into $C$ degrees Celsius. (a) Find $C$ when $F = 100$ . Give your answer correct to 3 significant figures. (b) Find $C$ when $F = - 4$ .

Show the working

(a) Substitute $F = 100$ . Work out the bracket first.
$C = \frac{5 (100 - 32)}{9} = \frac{5 \times 68}{9} = \frac{340}{9}$
Divide: $340 \div 9 = 37.777 \dots$ . Round only now, to 3 significant figures: $37.8$ .
On a calculator, type $5 \times (100 - 32) \div 9$ . Without the bracket, the calculator would work out $5 \times 100 - 32 \div 9$ , which is a different calculation.
(b) Substitute $F = - 4$ in brackets. The bracket in the formula is now $(- 4) - 32 = - 36$ .
$C = \frac{5 \times (- 36)}{9} = \frac{- 180}{9} = - 20$
Check the sign: $- 4$ °F is below $32$ °F, the temperature at which water freezes, so a Celsius value below $0$ makes sense.

Answer: (a) $37.8$ °C (3 s.f.) (b) $- 20$ °C

Where people go wrong

Mistake: Writing $3 a$ as $34$ when $a = 4$ .

Instead: $3 a$ means $3 \times a$ , so $3 a = 3 \times 4 = 12$ .

Algebra hides the multiplication sign. Writing the digits side by side makes a new two-digit number, and nothing is multiplied.

Mistake: Working out $2 x^{2}$ as ${(2 x)}^{2}$ , e.g. getting $36$ when $x = 3$ .

Instead: Square first, then multiply: $2 x^{2} = 2 \times 3^{2} = 2 \times 9 = 18$ .

The power belongs only to the $x$ , and the order of operations does powers before multiplication.

Mistake: Getting $x^{2} = - 9$ when $x = - 3$ , by typing $- 3^{2}$ into a calculator.

Instead: Use brackets: ${(- 3)}^{2} = 9$ .

Without brackets the calculator squares the $3$ and then applies the minus sign. A negative number squared is always positive.

Mistake: Reading $p$ as 'a pen', so that $4 p$ means 'four pens'.

Instead: Say what the letter stands for as a number: ' $p$ is the price of one pen in dollars'. Then $4 p$ is the cost of four pens, in dollars.

A letter always stands for a number. If you treat it as an object, you cannot substitute a value into it or check that your expression is right.

Mistake: Typing $\frac{a + b}{2 c}$ into a calculator as $a + b \div 2 \times c$ .

Instead: Type $(a + b) \div (2 \times c)$ , with brackets round the whole top and the whole bottom.

The calculator cannot see a fraction line written on paper. Without brackets it divides only $b$ by $2$ , multiplies that by $c$ , and then adds $a$ .

Check yourself

Write an expression for the number that is $5$ less than $y$ .
Answer
$y - 5$ . (Not $5 - y$ : that is $y$ less than $5$ .)
A bag holds $b$ sweets. Write an expression for the number of sweets in $6$ bags after $4$ of the sweets have been eaten.
Answer
$6 b - 4$
Is $a - b = b - a$ true for every pair of numbers $a$ and $b$ ? Use numbers to decide.
Answer
No. With $a = 5$ and $b = 2$ , $a - b = 3$ but $b - a = - 3$ . One pair of numbers that fails is enough to show that it is not true in general. (It is true only when $a = b$ .)
Explain the difference between $3 n$ and $n^{3}$ . Then find the value of each when $n = 2$ .
Answer
$3 n = 3 \times n$ , but $n^{3} = n \times n \times n$ . When $n = 2$ , $3 n = 6$ and $n^{3} = 8$ .
Without a calculator, find the value of $x^{2} - 4 x$ when $x = - 2$ .
Answer
${(- 2)}^{2} - 4 \times (- 2) = 4 + 8 = 12$
Find the value of $3 p^{2} q$ when $p = 5$ and $q = - 1$ .
Answer
$3 \times 5^{2} \times (- 1) = 3 \times 25 \times (- 1) = - 75$
The formula $s = \frac{(u + v) t}{2}$ gives a distance $s$ . Find $s$ when $u = 4$ , $v = 10$ and $t = 5$ .
Answer
$s = \frac{(4 + 10) \times 5}{2} = \frac{70}{2} = 35$
Use the formula $v = u + a t$ to find $t$ when $v = 25$ , $u = 5$ and $a = 4$ .
Answer
$25 = 5 + 4 t$ , so $20 = 4 t$ and $t = 5$ .

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. In this question, $a = - 2$ , $b = 5$ and $c = 3$ . Do not use a calculator.

(a) Work out the value of $4 a + b c$ . 2 marks No calculator

Mark scheme

M1 4 × (−2) + 5 × 3 oe, or −8 and 15 both seen
A1 7 cao

(b) Find the value of $a^{2} - 2 b$ . 2 marks No calculator

Mark scheme

M1 (−2)² − 2 × 5 oe, or 4 − 10 seen
A1 −6 cao

(c) Find the value of $\frac{a^{3} + b}{c}$ . 2 marks No calculator

Mark scheme

M1 ((−2)³ + 5) ÷ 3 oe, or a numerator of −3 seen
A1 −1 cao

Question 2. Adult tickets for a concert cost $x$ dollars each. Child tickets cost $y$ dollars each.

(a) Write an expression for the total cost, in dollars, of $3$ adult tickets and $5$ child tickets. 1 mark No calculator

Mark scheme

B1 3x + 5y oe

(b) A family buys $2$ adult tickets and $3$ child tickets. They pay with a $$ 100$ note. Write an expression for their change, in dollars. 2 marks No calculator

Mark scheme

B2 100 − 2x − 3y oe, e.g. 100 − (2x + 3y). If B0, award B1 for 2x + 3y seen

(c) Work out their change when $x = 24.50$ and $y = 13.75$ . 2 marks No calculator

Mark scheme

M1 100 − 2 × 24.50 − 3 × 13.75 oe; FT their (b)
A1 9.75 cao

Question 3. A company hires out vans. The cost, $C$ dollars, of hiring a van for $d$ days and driving it $m$ kilometres is given by the formula $C = 45 d + 0.32 m$ .

(a) Calculate $C$ when $d = 3$ and $m = 418$ . 2 marks

Mark scheme

M1 45 × 3 + 0.32 × 418 oe
A1 268.76 cao

(b) Hana hires a van for $4$ days. Her cost is $$ 295.20$ . Calculate the number of kilometres that she drives. 3 marks

Mark scheme

M1 295.20 = 45 × 4 + 0.32m oe
M1 0.32m = 115.2 or m = (295.20 − 180) ÷ 0.32 oe
A1 360 cao

Question 4. An object of mass $m$ kg moves at a speed of $v$ m/s. Its kinetic energy, $E$ joules, is given by the formula $E = \frac{1}{2} m v^{2}$ .

(a) Calculate $E$ when $m = 0.43$ and $v = 12.7$ . Give your answer correct to 3 significant figures. 2 marks

Mark scheme

M1 0.5 × 0.43 × 12.7² oe
A1 34.7 (34.677…) awrt

(b) The speed of the object is doubled. Its mass does not change. Explain what happens to $E$ . 2 marks

Mark scheme

B1 E is multiplied by 4 oe, e.g. E becomes four times as big
B1 a reason from the formula, e.g. ½m(2v)² = 4 × ½mv², or v is squared so the factor is 2² = 4

How this is examined

Finding a letter that is not the subject of a given formula: A Calculate part that gives a formula and the values of every letter except one, and that one is not the subject The known values substituted into the formula as it is printed, e.g. $295.20 = 45 \times 4 + 0.32 m$ (M1). The equation that is left solved one step at a time, e.g. $0.32 m = 115.2$ (M1). The value, $m = 360$ (A1). Write the substituted line before you rearrange anything.
Substituting a negative number into an expression with a power, without a calculator: A Find the value part, on the non-calculator paper, for an expression such as $a^{2} - 2 b$ or $a b^{2}$ when one letter is negative The substitution written with brackets round each negative, e.g. ${(- 2)}^{2} - 2 \times 5$ (M1), then the correct value, $- 6$ (A1). ${(- 2)}^{2}$ is $4$ , and ${(- 2)}^{3}$ is $- 8$ .
Using a given formula with decimal values and rounding the answer to 3 significant figures: A two-mark Calculate part on the calculator paper, with the formula printed in the question The numbers in the formula, seen, e.g. $0.5 \times 0.43 \times {12.7}^{2}$ (M1). Then the full value, $34.677 \dots$ , rounded once to $34.7$ (A1). If the answer is exact to 4 or 5 significant figures, such as $268.76$ , give it in full.
Writing an expression from a description in words: A Write part that asks for a total cost, an amount of change or a number of items, with letters for the unknown amounts The exact expression, with the number in front of each letter and a bracket when a whole amount is taken away: $100 - (2 x + 3 y)$ or $100 - 2 x - 3 y$ . No equals sign and no words inside the expression. $100 - 2 x + 3 y$ is a different expression.
Explaining what is true for every value of a letter: An Explain or Show (that) part about a formula, or about any whole number $n$ Algebra that works for every value, e.g. ${(2 v)}^{2} = 4 v^{2}$ or $n + (n + 1) = 2 n + 1$ , and then a sentence that states the conclusion, e.g. ' $E$ is multiplied by $4$ '. A check with one or two numbers supports your answer, but it does not show that the result is true for every value.

Words to know

Generalised number: A letter that stands for any number, not one particular number. A statement such as $a + b = b + a$ uses generalised numbers and is true for every value. A statement such as $a - b = b - a$ also uses them, but it is not true for every value.
Variable: A letter that stands for a number that can change or that is not yet known.
Expression: Letters and numbers joined by operations, with no equals sign, e.g. $4 p + q$ .
Term: One part of an expression, separated from the other parts by $+$ or $-$ , e.g. $4 p$ in $4 p + q$ .
Coefficient: The number that multiplies a letter in a term. In $- 2 y$ the coefficient of $y$ is $- 2$ .
Formula: An equation that links quantities, e.g. $v = u + a t$ .
Subject: The letter on its own on one side of a formula. In $v = u + a t$ the subject is $v$ .
Substitute: Replace each letter with a given number, and then work out the value.

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