2.9.1 Forming equations
An equation is made up of two expressions that are separated by an equal sign.
e.g.
\( 4+5=3^2 \)
\( l=3×w \)
To form an equation from the given information
You must first create an expression that is equal to a value or another expression
Define a variable
Write the equation using the variable
It helps to be aware of alternative terms for fundamental operations
Addition
Sum, total, more than, increase, etc.
Subtraction
Difference, less than, decrease, etc.
Multiplication
Product, lots of, times as many, double, triple, etc.
Division
Shared, split, grouped, halved, quartered, etc.
Worked example:
Oranges cost \( 21 \) cents each.
Alex buys \( x \) oranges and Bobbie buys \( (x+2) \) oranges.
The total cost of these oranges is \( $ \ 4.20 \).
Find the value of \( x \).
The cost of an orange in dollars
\( \frac{21}{100}=0.21 \)
The cost of Alex’s oranges
\( 0.21x \)
The cost of Bobbie’s oranges
\( 0.21(x+2) \)
The cost of all the oranges that are bought
\( 0.21x+0.21(x+2)=4.20 \)
Expand the brackets and collect like terms
\( 0.21x+0.21x+0.42=4.20 \)
\( 0.42x+0.42=4.20 \)
\( 0.42x=4.20-0.42 \)
Solve for \( x \)
\( 0.42x=3.78 \)
\( x=9 \)
\( x=9 \)
2.9.1.1 Forming equations from shapes
To form an equation involving the area or perimeter of a two-dimensional shape
Examine the question carefully to determine whether it involves area or perimeter
If no diagram is provided, it is almost always a good idea to draw one quickly
Add any information from the question to the diagram
If the question is about perimeter, determine which sides are equal in length and add them together
If the question involves area, write down the formula for that shape’s area
Worked example:
In this part, all measurements are in meters.
The diagram shows a rectangle.
The area of the rectangle is \( 310 \)\( m^2 \)
Work out the value of \( w \).
As the given shape is a rectangle, the parallel sides are equal
\( 5x-9=3x+7 \)
Solve for \( x \)
\( 5x-3x=7+9 \)
\( 2x=16 \)
\( x=8 \)
Substitute \( x=8 \) in \( 3x+7 \) or \( 5x-9 \) to get the length of the rectangle
\( 5(8)-9=31 \)
As area of the rectangle is given by \( Area=Length\times Width \)
Substitute the value of area and the value of length
\( 310=31\times w \)
\( w=\frac{310}{31} \)
\( w=10 \)
\( w=10 \)
To form an equation involving the angles in a two-dimensional shape
Examine the question carefully to determine whether it involves angles
If no diagram is provided, it is almost always a good idea to draw one quickly
Add any information given in the question to the diagram
Use angle properties
Look for key information that can provide additional information about the angles
To form an equation involving the surface area or volume of a three-dimensional shape
Examine the question carefully to determine whether it involves the surface area or volume
If no diagram is provided, it is almost always a good idea to draw one quickly
Add any information given in the question to the diagram
Write down the necessary formula
Worked example:
In the diagram, \( K \), \( L \) and \( M \) lie on a circle, center \( O \).
Angle \( KML=2x° \) and reflex angle \( KOL=11x° \) .
Find the value of \( x \).
As the circle theorem states that the angle at center of the circle is
twice the angle at circumference
Angle \( KOL=2KML \)
Angle \( KOL=2(2x°)=4x° \)
Put this information on to the diagram
Angles around a point add up to \( 360° \) .
\( 11x+4x=360 \)
Solve for \( x \)
\( 15x=360 \)
\( x=\frac{360}{15} \)
\( x=24 \)
\( x=24 \)
Test yourself:
Question 1:
Ahmed sells different types of cake in his shop.
The cost of each cake depends on its type and its size.
Every small cake costs \( $ \ x \) and every large cake costs \( $ \ (2x+1) \).
The total cost of \( 3 \) small lemon cakes and \( 2 \) large lemon cakes is \( $12.36 \).
Find the cost of a small lemon cake.
[3]
The cost of \( 18 \) small chocolate cakes is the same as the cost of \( 7 \) large chocolate cakes.
Find the cost of a small chocolate cake.
[3]
Question 2:
The angles of a quadrilateral are \( x° \), \( (x+5)° \), \( (2x-25)° \) and \( (x+10)° \).
Find the value of \( x \).
[3]
Question 3:
Factorise \( y^2+5y-84 \).
[2]
The area of the rectangle is \( 84 \) \( cm^2 \).
Find the perimeter.
[2]