Exemplification
Example A
 (i) The first five terms of an arithmetic sequence are
Find, in terms of n, an expression for the nth term of the sequence.
Jane says that 697 is a term in the arithmetic sequence,
(ii) Is Jane correct?
You must justify your answer.
 The nth term of a different number sequence is 4n + 5
Work out the first three terms of this number sequence.
 Here are some patterns made from matchsticks:



Pattern number 1 
Pattern number 2 
Pattern number 3 
(i) Draw Pattern number 4.
(ii) Complete this table for the pattern sequence
Pattern number 
1 
2 
3 
4 
5 

… 
100 
Number of matchsticks 
4 
7 
10 





(iii) Find, in terms of n, an expression for the nth term of the sequence
Example B
 Eggs are sold in boxes.
A small box holds 6 eggs.
Hina buys x small boxes of eggs.
Write down, in terms of x, the total number of eggs in these small boxes.
 Mr Smith owns minibuses and coaches.
Each minibus has 12 seats.
(i) Write an expression, in terms of m, for the number of seats in m minibuses.
Each coach has 48 seats
(ii) Write an expression, in terms of
m and c, for the number of seats in m minibuses and c coaches.
Example C
Simplify these expressions:
 3a + 2b + 2a – b
 4x + 7 + 3x – 3 – x
 3(x + 5)
 12 – (n – 3)
 m(n – p)
 4(a + 2b) – 2 (2a + b)
 3(x – 2) – 2(4 – 3x)
 (n + 1) 2 – (n + 1) + 1
Factorise:
 4a + 10b
 p^{2} + 6p
 6x^{2}  9xy
Example D
 Solve the equation 5x = 30
 Solve x + 2x = 12
 Solve 2y – 1 = 13
 Solve 4(x + 3) = 6
Example E
 P = 3a + 5b
a = 5.8
b = –3.4
Work out the value of P.
 S = 2p + 3q
p = – 4
q = 5
(i) Work out the value of S.
T = 2m + 30
T = 40
(ii) Work out the value of m.
 P = x^{2}  7x
Work out the value of P when x =  5
 P = 𝜋r + 2r + 2a
P = 84
r = 6.7
Work out the value of a.
Give your answer correct to 3 significant figures.
Example F
 Expand and simplify (x + 7)(x – 4)
 Expand y(y^{3} + 2y)
 Expand p(q – p^{2})
 Expand and simplify 5(3p + 2) – 2(5p – 3)
Example G
 Rearrange the equation y = 4x  5 to find x in terms of y
 Make t the subject of the formula v = 5t + u
 Make c the subject of the formula a = 3c – 4
Example H
Harry left school at 3 30 pm.
He walked home.
On the way home, he stopped to talk to a friend.
His sister, Sophie, left the same school at 3 45 pm.
She cycled home using the same route as Harry.
Here are the distancetime graphs for Harry’s and Sophie’s complete journeys.
 Find the distance Harry walked during the first 10 minutes of his journey.
 Find the total time that Harry stopped to talk to his friend.
 Write down the distance that Harry had walked when Sophie cycled past him.
Example I
Here are five graphs labelled A, B, C, D and E.
Each of the equations in the table represents one of the graphs A to E.
Write the letter of each graph in the correct place in the table.
Equation 
Graph 
x + y = 5 

y = x  5 

y = 5  x 

y = 5 

x = 5 

Example J
Diagram NOT accurately drawn
The diagram shows two straight lines intersecting at point A.
The equations of the lines are
y = 4x – 8
y = 2x + 3
Work out the coordinates of A.
 Solve the simultaneous equations
5a + 3b = 9
2a – 3b = 12
 Solve the simultaneous equations
6x – 2y = 33
4x + 3y = 9
 The graphs of the straight lines with equations 3y + 2x = 12 and y = x – 1 have been drawn on the grid.
Use the graphs to solve the simultaneous equations
3y + 2x = 12
y = x –1
Example K
 A straight line passes through the points (0, 5) and (3, 17).
Find the equation of the straight line.
 Find the gradient of the straight line with equation 5y = 3 – 2x.