Orienta

tion

You should be able to

■ solve simple equations1 What value must the � represent in each case?

a � � 3 � 12 b � � 5 � 20 c � � 11 � 19 d �2 � 100

■ interpret algebraic expressions2 What do each of these expressions mean? For example, 5x means that ‘I think of a number and I multiply it by 5’.

a 6y b p2 c 2m � 3 d 4(h � 2)

e t __ 7 f 2x2 g (2x)2 h 10 � 3b

■ expand brackets and simplify3 Expand these brackets

a 3(x � 9) b 4(2x � 1) c �3(4y � 2) d x(x � 7)e (x � 5) (x � 2) f (y � 7) (y � 3) g (w � 4)2 h (y � 6) (y � 9)

■ factorise quadratic expressions4 Factorise into double brackets

a x2 � 5x � 6 b x2 � 2x � 24 c x2 � 6x � 9 d x2 � 100

Check in

Ric

h t

ask In this diagram the equation

3x � 2 � 17 has been changed in different ways but all of these ways still give the same solution of x � 5.Describe each change to the equation.Continue each change for at least one more step.Invent some changes of your own.

A5 Solve simultaneous equations

A6 Plot and interpret further equations

What I need to know What I will learn What this leads to

A-level

Maths, Chemistry, Physics

■ Use inverse operations with algebra

■ Solve linear equations including those with fractions and brackets

■ Solve quadratic equations by factorisation

A2 Expand brackets

KS3 Experience of solving equations

EquationsMathematicians use algebra to turn real world problems into mathematical equations. Formula 1 engineers use complex mathematical equations to predict the effect on performance of their cars when they make technical modifications.

What’s the point?Learning how to write and solve equations allows you to understand the real world and make predictions.

A4

Equatio

ns

1.5x � 1 � 8.5

6x � 4 � 34

3x � 3 � 18

4x � 2 � x � 172x � 2 � 17 � x

3x � 2 � 17

3x � 1 � 16

Probability revision

282282 283283

Unit

3

Working with inverse operationsA4.1Keywords

InverseOperationVariable

● Addition, subtraction, multiplication and division are all operations.

● An operation has an inverse, for example subtraction is the inverse of addition.

This spread will show you how to:● Understand and use inverse operations● Use function machines

An operation acts on a number or a variable.

The inverse operation changes the variable back to its original value.

a� 3a � 33a

If two or more operations act on a variable then you do their inverse operations in reverse order to get back to the original value.

� 5

� 5

� 2

� 2

x 2x

x 2x

2x � 5

2x � 5

Grade DExercise A4.1

x and y are examples of variables.

The operations are ‘multiply by 2’ and ‘subtract 5’.

The inverse operations are ‘add 5’ and ‘divide by 2’.

To find p, do the inverse operations in the reverse order.

Undo the operations in the reverse order.

Exa

mple

p Find the value of each letter.

�7

xyz

53

a 5b �12cd p

12

�3

a 5 →�3

15 →�7

8 So x � 8

b �12 →�3

�36 →�7

�43 So y � �43

c 1 _ 2 →

�3 1 1 _

2 →

�7 �5 1 _

2 So z � �5 1 _

2

d 53 →�7

60 →�3

20 So p � 20

Exa

mple

p Find the starting number in each case.

a I think of a number, subtract 8 and square root it. The answer is 5.b I think of a number, multiply by 2, subtract 4 and cube it. The answer

is �216.

a 5 → 25 →�8

25 � 8 � 33 The starting number is 33.

b �216 → �6 →�4

�2 →�2

�1 The starting number is �1.

square

cube root

1 Copy and complete these diagrams.

a

� 5

15

20�8

x� 4

b

� 1.51.5

13.5 y

5

Square

c 5�7

12

� 2

18

w

d 5�2

� 21282000

Cube

3 In each case, use inverse operations to find the starting number.a I think of a number, double it and subtract 4. This gives me 7.b I think of a number, square it, multiply by 3 and get 75.c I think of a number, add 11, cube root it and get 2.d I think of a number, halve it, treble it then subtract 6. I get 54.e I think of a number, multiply it by 1 _

4 , square root it and get 5.

4 Is there another starting number for question 3b? Explain your answer.

5 a Write, in order, the operations that have acted on x.

i x � 3 � 17 ii 2x � 1 � 17 iii x � 5 _____ 2

� 24 iv 3x2 � 48

b Using the inverse operations, in reverse order, find the value of x in each case.

6 Look at this function machine.

Start Finish� 9 � 5 � 2 �� 2 12

a What value do you end up with if (�3) enters the machine?b What value did you start with if these values leave the machine? i 6 ii 8.4 iii 1 _

4

7 a Invent your own function machine that uses five operations to convert a value of 5 into 53.

b Use your machine to describe what value you started with if the output is 100.

AO

3P

roble

m 2 Explain why ‘adding 10%’ cannot be undone using the inverse operation ‘subtracting 10%’.

Probability revision

284284 285285

Unit

3

Solving one-sided equationsA4.2Keywords

EquationOperationSolve

This spread will show you how to:● Set up and solve simple equations

Grade CExercise A4.2Exa

mple

p Solve these equations.

a 3x � 5 _______ 2 � 8 b 4(y � 3) � 40 c 17x � 35

a 3x � 5 _______ 2 � 8 Multiply both sides by 2.

3x � 5 � 16 Add 5 to both sides. 3x � 21 Divide both sides by 3. x � 7

b 4(y � 3) � 40 Divide both sides by 4. or 4(y � 3) � 40 Expand the bracket. y � 3 � 10 Subtract 3 from both sides. 4y � 12 � 40 Subtract 12 from both sides. y � 7 4y � 28 Divide both sides by 7. y � 7

c 17x � 35 Divide both sides by 17.

x � 35 __

17 Change to a mixed number.

� 2 1 __ 17

Avoid decimals. 1 __ 17

is a recurring decimal and has to be rounded, so a decimal answer is not exact.

Exa

mple

p The perimeter of this rectangle is 34 cm. Find x.

3x�1

x

perimeter � x � x � 3x � 1 � 3x � 1 � 8x � 2 so 8x � 2 � 34 8x � 32 x � 4

This equation is only true when x � 11.

Undo the operations in reverse order.

The interior angle sum of a quadrilateral is 360º.

● An equation is a statement with an equals sign. For example 2x � 4 � 18 is an equation.

● To solve an equation, do the same operation to both sides. For example

2x � 4 � 18 Add 4 to both sides. 2x � 22 Divide both sides by 2. x � 11

AO

3P

roble

m 6 The diagram shows an isosceles triangle. Given that the equal angles are 10 less than double the third angle:a Write an expression for each equal angle.b Write an equation connecting the angles.c Solve the equation, using your answer to find the size of

each angle.

x

1 Solve these one-sided equations.a 3x � 7 � 8 b 4(x � 1) � 20 c x __

2 � 8 � 13

d 2x � 8 _______ 4

� �3 e 2(x2 � 9) � 68 f 2 ( x � 1 _____ 2

� 3 ) � 8

g 3x � 5 _______ 2

� 10 h x3 � 4 � 12 i √ __ x � 1 � 9

2 Copy and complete this crossword by solving the equations in the clues.

1 2

3

4 5

6

Across

1 2(x � 2) � 30

3 � 5 � 15

4 � 71

6 3(x � 100) � 675

x23x � 11

2

Down

1 3x � 15 � 330

2 x � 10 � 30

5 x2 � 1 � 170

3 The perimeter of this shape is 30 mm. Find the length of each side.

5 � x

2x � 6

4 This shape is a quadrilateral. Work out the value of x.

66˚

3x

x

5 a Given that y � 4x � 8, work out the value of x when y � 12.b Repeat for y � �4.

p.232

Probability revision

286286 287287

Unit

3

Grade CExercise A4.3Solving double-sided equationsA4.3Keywords

One-sidedUnknown

This spread will show you how to:● Set up and solve simple equations

When you solve an equation with unknowns on both sides

Subtract the smaller algebraic term from both sides, for example 5x � 3 � 3x � 1 Subtract 3x from both sides. 2x � 3 � 1 This is now a one-sided equation.

Solve the one-sided equation by using inverse operations. 2x � 3 � 1 Add 3 to both sides. 2x � 4 Divide both sides by 2.

x � 2

Exa

mple

p Solve

a 3x � 7 � 5x � 1 b 5 � 6y � 2y � 3

a 3x � 7 � 5x � 1 Subtract 3x from both sides (3x is smaller than 5x). 7 � 2x � 1 Add 1 to both sides. 8 � 2x Divide both sides by 2. 4 � x x � 4

b 5 � 6y � 2y � 3 Subtract �6y from both sides (�6y is smaller than 2y): Subtracting �6y is the same as adding 6y.

5 � 8y � 3 Add 3 to both sides. 8 � 8y Divide both sides by 8. y � 1

0 2y�6y

Exa

mple

p The diagram shows an isosceles triangle.Find the length of the equal sides.

2z � 4 � 4z � 18 Write an equation and solve it. �4 � 2z � 18 After subtracting 2z from both sides. 14 � 2z After adding 18 to both sides. 7 � z

Hence, 2z � 4 � 14 � 4 � 10 and 4z � 18 � 28 � 18 � 10The equal sides of the triangle are 10 units long.

2z � 4 4z � 18

AO

3P

roble

m 4 In each case, use the information to form an equation and solve it.a The angles in a triangle are x°, x � 20° and x � 40°. Find the angles of the triangle.b The perimeters of these two shapes are equal. What are the dimensions of each shape?

4x � 3 4x � 1

8x

c Abdul is 180 cm tall and Mark is 10x cm tall.

Alice is 164 cm tall and Miranda is 9x cm tall.

The difference in height between the two boys is equal to the difference in height between the two girls.

How tall are Mark and Miranda?

x

x � 40˚

x � 20˚

Abdul Alice Mark Miranda

180 cm164 cm 10x 9x

1 Solve these equationsa 2x � 4 � x � 3 b 10 � 3x � 7x � 10c 8 � 3x � 5 � 2x d 4(7 � 2z) � 15 � 8z

2 In each row of equations, one has a different solution from the other two. Find the odd one out.a 2a � 7 � 4a � 1 6a � 2 � 2a � 6 10 � 2a � 7a � 5b 10 � 3b � 6b � 1 15 � 2b � 14 � b 3b � 14 � bc 2c � 2 � 4c � 1 8c � 7 � 6c � 4 1 � 4c � 2 � 8cd 2d � d � 8 � 3d � 2d � 4 5d � 7d � 3 � 10 � d 15 � d � d � 9 � d

3 In each case, use the information to write an equation and solve it to find the starting number.a I think of a number, multiply it by 8 and subtract 2. I get the same

answer as when I multiply this number by 2 and add 10.b I think of a number, multiply it by 5 and add 3. I get the same

answer as when I multiply it by 2 and subtract it from 24.c Taking double a number from 11 is equal to taking treble that

number from 14.

Probability revision

288288 289289

Unit

3

Solving equations with fractions Keywords

Cross multiplyReciprocal

This spread will show you how to:● Solve equations involving fractions

Grade BExercise A4.4

1 Solve these equations by cross multiplying.

a 7 __ x � 21 b 15 � 5 __ x c 4

__ y � 3 d 7 __ p � 8

e 10 ___ x � �2 f 11 � 5 __ y g �3 � 7 __ y h � 3 __ x � �9

2 Solve these equations

a 5 __ x � 9 � 10 b 10 ___ p � 7 � 8 c x __ 4

� 3 � 10 d �2 � 1 � 3 __ x

3 Solve these equations

a 16 ___ x � 4 � 2 b 12 ___ 2y

� 3 � 5 c 6 ___ 3p

� 1 � 10 d 15 ___ 2x

� 4 � �2

4 Solve these equations

a x � 1 _____ 3

� x � 1 _____ 4

b 2y � 1

_______ 3

� y __

2 c 5 ______

w � 5 � 15 ______

w � 7 d 3 _____

x � 1 � 9 _______

2x � 1

You can use the fact that x � 1 __ x � 1 to help you.

1 __ x is called the reciprocal of x.

● When you solve an equation involving fractions, clear the fractions first.

● Any non-zero number multiplied by its reciprocal is 1.

When the equation has just a single fraction on each side you can cross multiply to clear the fractions. For example,

� 3

715x

15 � 7 � x � 3 → 105 � 3x → x � 35 as above

Exa

mple

p I divide 15 by a certain number and get 3 _ 4 .

Write an equation and solve it to find the starting number.

If x is the missing number then: 15 ___ x � 3 __ 4 Cross multiply.

15 � 4 � 3x 60 � 3x, so x � 20

Cross multiply: LH numerator � RH denominator and vice versa.

Exa

mple

p Solve these equations

a 3 __ x � 4 __ 9 b

12 ___ p � 9 � 28

a 3 __ x � 4 __ 9 Simple fraction each

side, so cross multiply. b

12 ___ p � 9 � 28 Subtract 9 from

both sides.

3 � 9 � 4x Divide both sides by 4. 12 ___ p � 19 Cross multiply (19 � 19

__ 1 ).

27 � 4x 12 � 1 � 19p Divide both sides by 19.

x � 27 ___ 4 � 6 3 _

4 p � 12 ___

19

It is tempting to use cross multiplication straightaway but you cannot cross multiply until you have a simple fraction on each side.

Exa

mple

p Solve the equation 15 ___ x � 3 __ 7

15 ___ x � 3 __ 7 Multiply both sides by the common denominator 7x

to clear the fractions.

15 ___ x � 7x � 3 __ 7 � 7x Cancel common factors.

15 � 7 � 3 � x, so 105 � 3x, so x � 35

7x is the reciprocal of 1 ___

7x .

To find the mean add all the expressions and divide by the number of expressions.

Think of 21 as 21 ___

1

5 Write an equation and solve it to find the starting number in each case.a I think of a number, divide it into 16 and I get 10.b I think of a number, add 4, divide it into 12 and get 7.c I think of a number, take 3 and divide it into 11. This gives me the

same answer as when I take the same number and divide it into 8.

6 Use the formula P � 180 ______ n � 1

to find

a the value of P when n � 4

b the value of n when P � 12

c the value of n when P � 2 2 _ 9 .

p.386

AO

3P

roble

m 7 The means of each set of expressions are equal. Use this information to find x and, hence, the value of each expression.

2x � 1 3x � 2

6x � 45x � 4

10 � 2x7x

Set 1

3x � 7 5x � 8

13 � x

2(3x � 1) 12 � 4x

Set 2

p.28

A4.4

Probability revision

290290 291291

Unit

3

Grade CExercise A4.5Solving harder equationsA4.5Keywords

ExpandNegative term

This spread will show you how to:● Solve linear equations after simplifying them

When solving an equation with brackets, expand the brackets first.

4(x � 2) � 7(x � 1) Expand the brackets. 4x � 8 � 7x � 7 Subtract 4x from both sides. 8 � 3x � 7 Add 7 to both sides. 15 � 3x Divide both sides by 3. x � 5

An equation with a negative x-term is easier to solve if you get rid of the negative term first.

5 � 8x � 45 Add 8x to both sides to remove the �8x. 5 � 45 � 8x Subtract 45 from both sides. �40 � 8x Divide both sides by 8. x � �5

1 Solve these equationsa 4x � 9 � 15 b

y _

7 � 2 � 4

c 3x � 9 � 2x � 15 d 5p � 9 � 3p �7

e 10 � 3t � 8t � 12 f 3 � 7b � 9 � 10b

2 Solve these equations by first expanding brackets and/or collecting like terms.a 3x � 7x � 3 � 9 � 17 b 3(2y � 1) � 4(7y � 6)c 2(3 � 8z) � 4(5 � 6z) d 2a � 3(2a � 7) � 20e 3a � (9 � 7a) � 34 f 6 � (3x � 9) � �2(5 � x)g (x � 3)(x � 4) � (x � 7)(x � 2) h (y � 7)2 � (y � 5)2

3 Solve these equations containing negative terms.a 10 � 4x � 2 b 20 � 3y � 11 c 15 � 8z � 12d 14 � 7a � 19 e 18 � 17 � b f 36 � 2(3 � 6c)

Exa

mple

p The square and the rectangle have the same area.Find the value of x.

Area of square � Area of rectangle x2 � (x � 5)(x � 2) Remember brackets first (FOIL). x2 � x2 � 5x � 2x � 10 Collect like terms. x2 � x2 � 3x � 10 Subtract x 2 from each side. 0 � 3x � 10 Add 10 to both sides. 10 � 3x

10 __

3 � x

x � 3 1 _ 3

x x�5

x�2

Exa

mple

p Solve (x � 3)(x � 5) � (x � 2)2

(x � 3)(x � 5) � (x � 2)2

x2 � 3x � 5x � 15 � (x � 2)(x � 2) Remove brackets first (FOIL): multiply the terms. x2 � 8x � 15 � x2 � 2x � 2x � 4 Collect like terms. x2 � 8x � 15 � x2 � 4x � 4 Subtract x 2 from each side. 8x � 15 � 4x � 4 Subtract 4x from both sides. 4x � 15 � 4 Subtract 15 from both sides. 4x � �11

x � �11 ____

4 or �2 3 _

4

Don’t use a calculator – fractions are exact. Decimals that need rounding, make the solution less accurate.

AO

3P

roble

m 4 In each case, write an equation to represent the information given and solve the equation to find the starting number.a I think of a number, treble it and subtract this from 40.

My answer is 22.b I think of a number, add 6 and double the result. This gives me

the same answer as when I subtract 5 from the number and multiply the result by 6.

c I think of a number, add 7 and square it. This gives me the same answer as when I take 5 from the number and square it.

5 In each case, form an equation to represent the information given and solve it to find x.a

x � 11

x � 4

x � 3

The area of the square and rectangle are equal.

b

2x

4x

26

15

The lengths of the unmarked sides are equal.

p.186

Probability revision

292292 293293

Unit

3

Grade BExercise A4.6Introducing quadratic equationsA4.6Keywords

FactoriseProductQuadraticSolutionSum

This spread will show you how to:● Solve quadratic equations by factorisation

Many quadratic equations can be solved by factorising.

For example, solve the equation x2 � 5x � 6 � 0.

x2 � 5x � 6 � 0 The two factors will each start with x.(x � �)(x � �) � 0 Now find two numbers with a sum of 5 (for 5x) and a product of 6.(x � 3)(x � 2) � 0 Check; (x � 3)(x � 2) � x2 � 3x � 2x � 6 � x2 � 5x � 6.

The two factors have a product of zero. This means that at least one of them is equal to zero.

Either x � 3 � 0 or x � 2 � 0If x � 3 � 0, x � �3 and if x � 2 � 0, x � �2 Check (�3)2 � 5(�3) � 6 � 9 � 15 � 6 � 0The solutions are x � �3 and x � �2. and (�2)2 � 5(�2) � 6 � 4 � 10 � 6 � 0

x2 term, so equation is a quadratic.

Don’t forget to look out for common factors as well as double bracket style factorisation.

First make the left hand side of the equation equal to zero.

Exa

mple

p Solve these equations

a x2 � 7x � 12 � 0 b x2 � 3x � 0

a x2 � 7x � 12 � 0 To factorise, you need two negatives to multiply to make �12 and add to make �7. (x � 3)(x � 4) � 0

Either x � 3 � 0 or x � 4 � 0 x � 3 x � 4

b x2 � 3x � 0 x(x � 3) � 0 Either x � 0 or x � 3 � 0 x � �3

Exa

mple

p Solve x2 � 2x � 15

x2 � 2x � 15 � 0 Find two numbers with sum �2 and product �15.(x � 3)(x � 5) � 0 Either x � 3 � 0 or x � 5 � 0 x � �3 x � 5

● A quadratic equation contains an x 2 term as the highest power, for example x2 � 5x � 6 � 0

● The key to solving quadratic equations is that the product of the two factors is zero, so you can say that one or other (or both) of the factors is zero. This leads to the two solutions.

1 Here are six equations

x2� 100

5x � 3 � 3x x3� x � 10

x2� �49 x2� 169

2x2� 50

a Write the four quadratic equations.b From the quadratic equations, find three that have two solutions.

Write these three equations and their solutions.

2 Factorise these quadratic expressionsa x2 � 8x � 12 b x2 � 11x � 24 c x2 � 13x � 36d x2 � 16x � 55 e x2 � 7x � 12 f x2 � 10x � 24g x2 � 3x � 28 h x2 � 2x � 15

3 Solve these quadratic equations by factorising them into double brackets.a x2 � 7x � 12 � 0 b x2 � 8x � 12 � 0 c x2 � 10x � 25 � 0d x2 � 5x � 14 � 0 e x2 � 4x � 5 � 0 f x2 � 5x � 6 � 0g x2 � 10x � 21 � 0 h x2 � 3x � 40

4 Solve these quadratic equations by factorising them into a single bracket.a x2 � 8x � 0 b x2 � 4x � 0 c x2 � 6x � 0d y2 � 5y � 0 e x2 � 9x f x2 � 12x � 0g 2x2 � 8x � 0 h 6x � x2 � 0

5 Explain why x2 � 7x � 11 � 0 cannot be solved by factorisation.

6 a Vicky is trying to solve (x � 4)(x � 2) � 7. Here is her attempt.

What is wrong with her method?b Can you solve this equation correctly to

show that the two solutions should really be x � 3 and x � �5?

7 Simplify and then solve this equation x2 � 4x � 5 � 2x(x � 1)

(x + 4)(x – 2) = 7x + 4 = 7

x = 3x - 2= 7

x = 9oror

Either

p.192

294

Summary

Check out

295

Edexcel Limited, June 2008Edexcel Limited, June 2008

1 a Solve 6x � 7 � 38 (2)b Solve 4(5y � 2) � 40 (3)

(Edexcel Limited 2007)

2 a Solve 4y � 3 � 2y � 9 (2)

b Solve 5(t � 3) � 8 (2)(Edexcel Limited 2007)

3 Solve 6x � 5 � 2x � 9 (3)(Edexcel Limited 2006)

4 4x + 1

2x + 12

xx

The diagram shows a rectangle.All the measurements are in centimetres.

a Explain why 4x � 1 � 2x � 12 (1)

b Solve 4x � 1 � 2x � 12 (2)

c Use your answer to part b to work out the perimeter ofthe rectangle. (2)

(Edexcel Limited 2009)

5 a Factorise x2 � 6x � 8 (2)

b Solve x2 � 6x � 8 � 0 (1) (Edexcel Limited 2006)

Worked exam question

2x

3x

x!30

The diagram shows a triangle.The sizes of the angles, in degrees, are 3x, 2x and x � 30Work out the value of x. (3)

(Edexcel Limited 2008)

You should now be able to:

● Set up and solve simple equations● Solve linear equations where the unknown appears on either

side or on both sides of the equation● Solve linear equations after simplifying them● Solve quadratic equations by factorisation

A4 Exam questions

Diagram NOT accurately drawn

The angles in a triangle add to 180�.

3x � 2x � (x � 30) � 1803x � 2x � x � 30 � 180 6x � 30 � 180 6x � 180 � 30 6x � 150 x � 25�3x � 75�2x � 50�x � 30 � 55�

Form the equation

Show each line of working out.

Check:75� � 50� � 55� � 180�

Diagram NOT accurately drawn