## Learning Outcomes

If you work through this section you should be able to:

• Recognise simple and complex algebraic expressions.
• Expand brackets in expressions.
• Factorise expressions.
• Evaluate expressions to find their value.
• Form expressions to represent rules and relationships between values.
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This section requires some understanding of basic definitions and processes in algebra (see Introduction to algebra section).

You may also find the Equations section interesting.

We use algebra to abbreviate information without loss of clarity or accuracy, for example to represent a formula, and to generalise rules and relationships by using letters to represent values.

We combine letters and numbers and more complex terms and expressions using operators (+, −, ×, ÷, raising to a power, etc.) and these follow the normal rules of arithmetic with a few extra conventions.

The rest of this section will look at how to manipulate algebra to evaluate, and form, expressions.

Expanding brackets is also known as multiplying out.

We do this when an expression is part of a more complex expression enclosed in brackets and is intended to be multiplied by another term (a number and/or letters). Typically, the multiplication sign is not shown but assumed.

For example, 6(a + 5b − 2c) means 6 × (a + 5b − 2c).

It may be necessary to 'expand' the bracket by multiplying every term in the bracket by the term outside the bracket.

Remember: when multiplying, like signs make a plus and opposite signs make a minus (see the 'Mathematical operations' section). In particular, note that a minus times a minus makes a plus.

#### Examples

3(a +2b − 3c)
= 3 × a + 3 × 2b + 3 × (−3c)
= 3a + 6b − 9c

−x(5x − 2y + 3z)
= −5x2 + 2xy − 3xz

−2a(−a2 + 3a − 7)
= 2a3 − 6a2 + 14a

This is essentially the opposite of expanding brackets. It is often required to make an expression more compact, either to simplify future manipulation or to make evaluation easier.

Look out for factors (numbers or letters) that are common to each term and work out what is left when you take that factor out.

Remember basic arithmetical factorising. For example, 4 is a common factor of 4, 8 and 12, leaving 1, 2 and 3 respectively.

Also remember the use of powers when multiplying a letter by itself, for example a2 is a factor of a2, a3 and a5, leaving 1, a and a3 respectively.

#### Examples

3a + 6b − 9c = 3(a + 2b − 3c)

4ab − 6ac − 8a2 = 2a(2b − 3c − 4a)

Check that you have factorised correctly by mentally multiplying out the bracket of your answer.

This is where the letters in the expression are given particular values and the value of the expression as a whole is calculated. Very commonly, this is done when the expression is a formula.

For example, the formula to turn Centigrade temperature to Fahrenheit is:

C=$\frac{5\left(F-32\right)}{9}$

As with arithmetic calculations, the rule about the order of evaluation is BEDMAS:

• Brackets
• Exponents (powers)
• Division and Multiplication

#### Examples

When w = 2

3w + 4 = 3 × 2 + 4 = 6 + 4 = 10

When a = 2, b = 3, c = −1

b2 + (3a + 2c) = 32 + [3 × 2 + 2 × ( −1)] = 32 + [6 + ( −2)] = 9 + 4 = 13

(Multiplication before Addition within the Bracket, then the 32 evaluated (E), then the final Addition).

By using letters to stand for values, general rules and information about a system can be expressed very concisely. Then such an expression can be evaluated for a specific situation using the particular values, or may be used to find how one variable behaves as others are changed.

Typically, you will form an expression to summarise the information in a formula, or as part of forming an equation to solve for an unknown quantity.

#### Simple examples

1. My sister is two years older than I am. Let my age be x years.
Then my sister is (x + 2) years.
2. A grapefruit costs twice as much as an orange which is y pence.
Then the grapefruit is 2y pence.
3. A cake costs w pence and a bun costs z pence. What is the cost of 2 cakes and 3 buns?
(2w + 3z) pence.

#### Complex examples

1. The time for one oscillation of a simple pendulum is 2 pi times the square root of its length divided by the constant G. Let L be the length and T the time.
T=2$\pi \sqrt{L}{G}}$
2. To convert temperatures from Fahrenheit to Centigrade, you must subtract 32 from the Fahrenheit temperature, then divide that by 9, and multiply by 5. Using obvious letters:
C=$\frac{5\left(F-32\right)}{9}$
Note: the use of brackets to indicate the priority of the sequence of operations.
3. To convert from Centigrade to Fahrenheit, divide the Centigrade temperature by 5, then multiply by 9, and add 32.
F=$\frac{9}{5}$C+32

A. Express algebraically a number that is:

1) 2 more than a

2) 3 times a

3) a more than b

4) 4 less than b

5) ½ of b

6) c less than d

7) ¾ of 7 times x

8) 4 times (b less than c)

B. Express algebraically:

1) How many pence in £y?

2) What is the cost of 3 eggs if an egg costs x pence?

3) How many centimetres in x metres?

4) What is the next highest number after n?

5) What number is 5 more than 2x + 3?

6) A man earns £a and spends £b. How much does he save?

7) How much change do I get in pence from £3 after buying 6kg of sugar at x pence per kilo? Express your answer in pence.

8) A man is y years old.

• How old was he x years ago?
• How old will he be in z years time?

9) What is the total cost of m books at n pence each? Give your answer in:

• Pence
• £

10) A man pays income tax at the rate of r pence in the £ on all except the first £g of his income of £h. How much does he have left after tax?

11) In a test, n pupils scored b marks, and m scored c marks. What was:

• The total score?
• The average mark?

C. Express the following statements algebraically using the letters given:

1) Mr Green earns twice as much as Ms Black.

Use G for Mr Green and B for Ms Black.

2) John is three years older than Mary, who is two years younger than Bill.

Write two expressions to represent these two facts.

Use J for John, M for Mary and B for Bill.

3) The sum of the two numbers is twice their difference.

Use a and b for the two unknown numbers.

4) A man runs a certain distance and then walks ¼ of the same distance. Altogether he travels 5 miles.

Let the distance run be d miles.

5) There are 3 sons in a family, and each is 4 years older than the next. The total age is 48.

Use a to represent the age of the youngest son.

A. Express algebraically a number that is:

1) 2 more than a

a + 2

2) 3 times a

3a

3) a more than b

b + a

4) 4 less than b

b - 4

5) ½ of b

b/2

6) c less than d

d - c

7) ¾ of 7 times x

3/4 (7x) = 21x/4

8) 4 times (b less than c)

4 (c - b)

B. Express algebraically:

1) How many pence in £y?

100y

2) What is the cost of 3 eggs if an egg costs x pence?

3x

3) How many centimetres in x metres?

100x

4) What is the next highest number after n?

n + 1

5) What number is 5 more than 2x + 3?

2x + 3 + 5 = 2x + 8

6) A man earns £a and spends £b. How much does he save?

£a - £b

7) How much change in pence do I get from £3 after buying 6kg of sugar at x pence per kilo? Express your answer in pence.

300 - 6x

8) A man is y years old.

• How old was he x years ago?

y - x

• How old will he be in z years time?

y + z

9) What is the total cost of m books at n pence each? Give your answer in:

• Pence

mn

• £

mn/100

10) A man pays income tax at the rate of r pence in the £ on all except the first £g of his income of £i. How much does he have left after tax?

£i - r (i - g) / 100

11) In a test, n pupils scored b marks, and m scored c marks. What was:

• The total score?

nb + mc

• The average mark?

(nb + mc) / (m + n)

C. Express the following statements algebraically using the letters given:

1) Mr Green earns twice as much as Ms Black.

Use G for Mr Green and B for Ms Black.

G = 2B

2) John is three years older than Mary, who is two years younger than Bill.

Write two expressions to represent these two facts.

Use J for John, M for Mary and B for Bill.

J = M + 3

M = B - 2

3) The sum of the two numbers is twice their difference.

Use a and b for the two unknown numbers.

(a + b) = 2 (a - b)

4) A man runs a certain distance and then walks ¼ of the same distance. Altogether he travels 5 miles.

Let the distance run be d miles.

d + d/4 = 5

5) There are 3 sons in a family, and each is 4 years older than the next. The total age is 48.

Use a to represent the age of the youngest son.

a + (a + 4) + (a + 8) = 48

which you can write as 3a + 12 = 48