GCSE Maths · Higher · Chapter 15: Equations and inequalities

Linear and simultaneous equations, inequalities and trial and improvement

Grades 4–9About 90 minutesCalculator allowed94 marks

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Before you start

Grades 4–5

You will need these in every question below.

A1
Solve $4 x - 7 = 21$ .
[1 mark]
Answer

A2
Expand and simplify $3 (2 x - 5) - 2 (x - 4)$ .
[2 marks]
Answer
A3
Work out the value of $2 a^{2} - b$ when $a = - 3$ and $b = 5$ .
[1 mark]
Answer
A4
List the integers $n$ such that $- 2 < n \leq 3$ .
[1 mark]
Answer
A5
Work out the lowest common multiple of 6 and 4.
[1 mark]
Answer

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

Solve $5 (x - 2) = 3 x + 4$ .

Hint Expand the bracket, then collect the

x

terms on the side with more of them.

B1
Solve $4 (2 x + 3) = 5 x - 9$ .
[3 marks]
Answer

Example 2

Solve $\frac{x + 3}{4} = \frac{x - 1}{2}$ .

Hint Multiply both sides by 4, the lowest common multiple of the denominators.

B2
Solve $\frac{2 x - 1}{3} + \frac{x + 2}{4} = 2$ .
[3 marks]
Answer

Example 3

Solve the simultaneous equations $3 x + 2 y = 16$ and $x - 2 y = 0$ .

Hint The

y

terms have the same size and opposite signs, so add the equations.

B3
Solve the simultaneous equations
$4 x + 3 y = 62 x - 5 y = 16$
[3 marks]
Answer $x = \dots\dots$ $y = \dots\dots$

Example 4

Solve $7 - 2 x > 1$ .

Hint Add

2 x

to both sides so that the

x

term is positive. Then you never divide by a negative number.

B4
Solve $3 (x - 4) \geq 5 x - 18$ .
[3 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
Solve $2 (3 x - 1) - (x + 4) = 4 (x + 2)$ .
[3 marks]
Answer

C2
Solve $\frac{x}{3} + \frac{x}{5} = 16$ .
[2 marks]
Answer
C3
Solve the simultaneous equations
$y = 2 x - 53 x + 2 y = 18$
[3 marks]
Answer $x = \dots\dots$ $y = \dots\dots$
C4
The number line shows an inequality.
(a)
Write down the inequality shown.
[2 marks]
Answer
(b)
List the integers that satisfy the inequality.
[1 mark]
Answer
Total for C4: 3 marks
C5
Solve $- 5 \leq 2 x + 3 < 11$ .
[2 marks]
Answer
C6
3 adult tickets and 2 child tickets for a show cost £41. 2 adult tickets and 5 child tickets cost £47.50.
Work out the cost of one adult ticket and the cost of one child ticket.
[4 marks]
Answeradult £ $\dots\dots$ child £ $\dots\dots$
C7
The equation $x^{3} + 6 x = 100$ has a solution between 4 and 5.
Use trial and improvement to find this solution. Give your answer to 1 decimal place. You must show your working.
[4 marks]
Answer
C8
The region $R$ is shown on the grid. Write down the three inequalities that define $R$ .
[3 marks]
Answer
C9
$n$ is an integer. $3 n + 2 > 11$ and $2 n - 3 \leq 15$ .
Find all the possible values of $n$ .
[3 marks]
Answer

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
Ella solves $8 - 3 x > 2$ . She writes, " $- 3 x > - 6$ , so $x > 2$ ."
Explain what Ella has done wrong, and write down the correct solution.
[2 marks]
Answer

D2
The diagram shows a quadrilateral. The angles are given in degrees.
Work out the size of the largest angle.
[3 marks]
Not drawn accurately
Answer
D3
The perimeter of the rectangle is 50 cm.
Work out the area of the rectangle.
[4 marks]
Not drawn accurately
Answer
D4
Solve the simultaneous equations
$5 x - 4 y = 233 x + 6 y = - 3$
[4 marks]
Answer $x = \dots\dots$ $y = \dots\dots$
D5
The lines $y = 1$ , $x + y = 6$ and $y = 2 x$ are drawn on the grid.
List all the points with integer coordinates that satisfy all three of $y > 1$ , $x + y < 6$ and $y \leq 2 x$ .
[3 marks]
Answer
D6
Triangle $A B C$ is isosceles with $A B = A C$ . The lengths of the sides are given in centimetres. The perimeter of the triangle is 43 cm.
Work out the length of $B C$ .
[6 marks]
Not drawn accurately
Answer
D7
A cuboid has a square base of side $x$ cm. Its height is $(x + 3)$ cm. Its volume is 250 cm³.
(a)
Show that $x^{3} + 3 x^{2} = 250$ .
[2 marks]
(b)
Use trial and improvement to find the value of $x$ . Give your answer to 1 decimal place. You must show your working.
[4 marks]
Answer
Total for D7: 6 marks
D8
Solve $\frac{3 - 2 x}{4} \geq x - 6$ .
[3 marks]
Answer
D9
Solve $\frac{x + 4}{3} - \frac{x - 2}{5} = 4$ .
[3 marks]
Answer
D10
The line $y = m x + c$ passes through the points $(3, 5)$ and $(7, 17)$ .
Work out the value of $m$ and the value of $c$ .
[3 marks]
Answer $m = \dots\dots$ $c = \dots\dots$

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
Three numbers $a$ , $b$ and $c$ satisfy $a + b = 11$ , $b + c = 15$ and $a + c = 14$ .
Work out the values of $a$ , $b$ and $c$ .
[3 marks]
Hint 1 · What to try
Add all three equations together.
Hint 2 · The first line
$2 a + 2 b + 2 c = 40$ , so $a + b + c = 20$ .
Hint 3 · The full method
Take each equation away from 20: $c = 20 - 11 = 9$ , $a = 20 - 15 = 5$ , $b = 20 - 14 = 6$ .
Answer $a = \dots\dots$ $b = \dots\dots$ $c = \dots\dots$

E2
The solution of $4 (x - a) = x + 9$ is $x = 7$ .
Use this to solve the inequality $2 x + a < 5 x - 13$ .
[3 marks]
Hint 1 · What to try
Put $x = 7$ into the equation to find $a$ .
Hint 2 · The first line
$4 (7 - a) = 16$ , so $28 - 4 a = 16$ and $a = 3$ .
Hint 3 · The full method
$2 x + 3 < 5 x - 13$ gives $16 < 3 x$ , so $x > \frac{16}{3}$ .
Answer
E3
$x$ and $y$ are integers with $x \geq 1$ , $y \geq 2$ and $x + y \leq 7$ .
Work out the greatest possible value of $2 x + 3 y$ .
[3 marks]
Hint 1 · What to try
These are the inequalities of region $R$ in question C8. Only the points in $R$ are allowed.
Hint 2 · The first line
$2 x + 3 y = 2 (x + y) + y$ . Make $x + y$ as big as possible, then make $y$ as big as possible.
Hint 3 · The full method
$x + y = 7$ and $y = 6$ (so $x = 1$ ) give $2 \times 7 + 6 = 20$ .
Answer
E4
A two-digit number has digits that add up to 11. When the digits are swapped round, the number goes up by 27.
Work out the number.
[3 marks]
Hint 1 · What to try
Call the tens digit $a$ and the units digit $b$ . The number is $10 a + b$ .
Hint 2 · The first line
$a + b = 11$ and $(10 b + a) - (10 a + b) = 27$ , so $9 b - 9 a = 27$ and $b - a = 3$ .
Hint 3 · The full method
Adding $a + b = 11$ and $b - a = 3$ gives $2 b = 14$ , so $b = 7$ , $a = 4$ and the number is 47.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can solve linear equations with brackets, fractions and the unknown on both sides.
I can solve simultaneous equations by elimination and by substitution, including in a context.
I can solve linear inequalities, show them on a number line, and describe a region with inequalities.
I can use trial and improvement to find a solution to 1 decimal place.

Answers and mark scheme

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