Syllabus Edition

First teaching 2023

First exams 2025

Equations & Problem Solving (CIE IGCSE Maths: Core)

Revision Note

Test yourself

Author

Mark

Last updated

7 October 2024

Forming & Solving Equations

How do I form expressions from words?

You can turn common phrases into expressions
- Use x to represent an unknown value
  
  2 less than "something" $x - 2$
  
  Double "something" $2 x$
  
  5 lots of "something" $5 x$
  
  3 more than "something" $x + 3$
  
  Half of "something" $\frac{1}{2} x or \frac{x}{2}$
Common words indicating basic operations are:
- Addition: sum, total, more than, increase
- Subtraction: difference, less than, decrease
- Multiplication: product, lots of, times as many, double, triple
- Division: shared, split, grouped, halved, quartered
Brackets help keep the order correct
- "something" add 1, then multiplied by 3
  - $(x + 1) \times 3$ which simplifies to $3 (x + 1)$
- Compare this to "something" multiplied by 3, then add 1
  - $x \times 3 + 1$ which simplifies to $3 x + 1$
You may have to choose which unknown to call x
- If Adam is 10 years younger than Barry, then Barry is 10 years older than Adam
  - Either represent Adam's age as $x - 10$ and Barry's age as $x$
  - Or represent Adam's age as $x$ and Barry's age as $x + 10$
- If Adam's age is half of Barry's age, then Barry's age is double Adam's age
  - So if Adam's age is $x$ then Barry's age is $2 x$
  - This makes the algebra easier (rather than using $x$ for Barry's age and $\frac{1}{2} x$ for Adam's age)

How do I form equations?

An equation is a statement with an equals sign that can be solved
Try to put in the phrase "is equal to" to see where the equals goes
- Lisa's age is double Aisha's age and the sum of their ages is ("is equal to") 27
  - Represent Aisha's age as $x$ and Lisa's age is $2 x$
  - The equation is $2 x + x = 27$
- When solving, always give the answer in context
  - $3 x = 27$ so $x = 9$
  - In context: "Lisa is 18 years old and Aisha is 9 years old"
Sometimes you might have two unknown values (x and y)
- Use the information to form two simultaneous equations

Worked example

A flowerbed has flowers of three different colours: red, yellow and purple.
The number of yellow flowers is three times the number of red flowers.
The number of purple flowers is 5 more than the number of yellow flowers.

If the difference between the number of purple flowers and red flowers is 29, find the number of yellow flowers.

Let the number of red flowers be x

$x$ red flowers

Multiply this by 3 to get the number of yellow flowers

$3 x$ yellow flowers

Add 5 to the previous result to get the number of purple flowers

$3 x + 5$ purple flowers

Find the difference between the number of purple and red flowers (purple subtract red, as purple is larger)

$3 x + 5 - x$

Set the difference equal to 29

$3 x + 5 - x = 29$

Simplify the left-hand side (3x - x = 2x)

$2 x + 5 = 29$

Solve the equation (subtract 5 then divide by 2)

$\begin{array}{rcl} 2 x & = & 29 - 5 \\ 2 x & = & 24 \\ x & = & \frac{24}{2} \\ x & = & 12 \end{array}$

This is not the answer to the question asked
The number of yellow flowers is 3x so multiply this answer by 3

There are 36 yellow flowers

Forming Equations from Shapes

How do I form equations from shapes?

You need to use all the information given on the diagram and any specific properties of that shape
Common 2D shapes that you should know properties for are
- Triangles: equilateral, isosceles, scalene, right-angled
- Quadrilaterals: square, rectangle, kite, rhombus, parallelogram, trapezium
You may be asked about perimeter, area or angles
You may be asked about polygons
- Regular vs irregular polygons
- Interior vs exterior angles
  - The sum of interior angles is 180(n-2) for an n-sided polygon
You may be asked about angles in parallel lines
- Alternative, corresponding and co-interior
You may be asked about 3D shapes involving surface area and volume
- Prisms have constant cross sections
  - Volume is cross-section area multiplied by length

Is there anything else that can help?

Sketch a diagram if none is given
Split up uncommon shapes into the sum or difference of common shapes
Look out for important extra information
- For example, a trapezium "with a line of symmetry"
With irregular shapes, assume all angles and lengths are different (unless told otherwise)
Put brackets around algebraic expressions when substituting them into geometric properties

Forming and solving an equation from an irregular polygon

Examiner Tip

Read the question carefully - does it want an angle? perimeter? total area? curved surface area? etc.
For surface area and volume questions, check the list of formulas given in the exam.

Worked example

A rectangle has a length of $3 x + 1$ cm and a width of $2 x - 5$ cm.

Its perimeter is equal to 22 cm.

(a)

Use the above information to find the value of x.

The perimeter of a rectangle is 2 × length + 2 × width

$2 (3 x + 1) + 2 (2 x - 5)$

Expand the brackets

$6 x + 2 + 4 x - 10$

Simplify by collecting like terms

$10 x - 8$

This perimeter is 22, so set this expression equal to 22

$10 x - 8 = 22$

Solve this equation by adding 8 then dividing by 10

$\begin{array}{rcl} 10 x & = & 22 + 8 \\ 10 x & = & 30 \\ x & = & \frac{30}{10} \\ x & = & 3 \end{array}$

$x = 3$

(b)

Find the area of the rectangle.

The area of a rectangle is its length multiplied by is width
Substitute the value of x from part (a) into the length and width given in the question

length is 3 × 3 + 1 = 10

width is 2 × 3 - 5 = 1

Find the area (multiply length by width)

10 × 1

Include the correct units for area

Area = 10 cm²

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