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11. Coordinate Geometry
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iGCSE (2021 Edition)

11.08 Linear rules

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Identifying linear relationships

A relationship between two variables is linear if both of the following conditions are met:

  • a linear equation can be used to relate the two variables
  • the dependent variable changes by a constant amount as the independent variable changes

If we are given the graph of a relationship, it is very easy to see if it forms a straight line or not, but for now we will look at how to identify a linear relationship from either its table of values, or just from its equation.

From a table of values

When determining a relationship between two variables, a table of values can be used to display several values for a given independent variable ($$x) with corresponding values of the dependent variable ($$y).

A table of values makes it easy to identify if a relationship is linear or not. If there is a common difference between $$y values as $$x changes by a constant amount, then there is a linear relationship.

 

Worked Example

Does the following table of values represent a linear relationship?

$$x $$1 $$2 $$3 $$4 $$5
$$y $$8 $$16 $$24 $$32 $$40

Think: In a linear relationship, the $$y-value must change by equal amounts as the $$x-value increases by $$1. We can see that the $$x-values in this table of values are increasing by $$1 each step, so we want to find out if the $$y-values are changing by equal amounts each step.

Do: We can add an extra row to the bottom of our table of values to show the change in $$y-value at each step. We can see straight away that the $$y-value is increasing for each step, but by how much?

$$x $$1 $$2 $$3 $$4 $$5
$$y $$8 $$16 $$24 $$32 $$40


We can see that the $$y-value always increases by $$8 as the $$x-value increases by $$1. This means the table of values does represent a linear relationship.

Reflect: By comparing the increases in the $$y-value as the $$x-value increases by $$1, we can determine if an equation is linear. 

 

Careful!

The $$x-values in a table of values might not necessarily increase by $$1 each step. However, we can still use this method by dividing the increase or decrease in the $$y-value by the increase in the $$x-value to find the unit change.

Practice question

Question 1

Consider the relationship between $$x and $$y in the table below.

$$x $$1 $$2 $$3 $$4 $$5
$$y $$5 $$1 $$−3 $$−7 $$−11

  1. Is the relationship linear?

    Yes, the relationship is linear.

    A

    No, the relationship is not linear.

    B

Writing rules for relationships

When constructing a linear equation from a worded sentence, look for terms such as sum, minus, times, and equals. We can convert the description into a linear equation by using mathematical symbols in the place of words.

 

Linear equations 

All linear relationships can be expressed in the form: $$y=mx+c.

  • $$m is equal to the change in the $$y-values for every increase in the $$x-value by $$1.
  • $$c is the value of $$y when $$x=0.

 

Practice question

Question 2

Consider the relationship between $$x and $$y in the table below.

$$x $$1 $$2 $$3 $$4 $$5
$$y $$6 $$12 $$18 $$24 $$30

  1. Which of the following options describes the relationship between $$x and $$y?

    The $$y-value is equal to the $$x-value plus five.

     

    A

    The $$y-value is equal to five times the $$x-value.

    B

    The $$y-value is equal to six times the $$x-value.

    C

    The $$y-value is equal to the $$x-value.

    D

  2. Write the linear equation that describes this relationship between $$x and $$y.

Question 3

The variables $$x and $$y are related, and a table of values is given below:

$$x $$1 $$2 $$3 $$4 $$5
$$y $$−2 $$−4 $$−6 $$−8 $$−10

  1. What is the value of $$y when $$x=0?

  2. Write the linear equation expressing the relationship between $$x and $$y.

  3. What is the value of $$y when $$x=−16?

Question 4

The variables $$x and $$y are related, and a table of values is given below:

$$x $$0 $$1 $$2 $$3 $$4 $$5
$$y $$8 $$13 $$18 $$23 $$28 $$33

  1. Linear relations can be written in the form $$y=mx+c.

    For this relationship, state the values of $$m and $$c:

    $$m=

    $$c=

  2. Write the linear equation expressing the relationship between $$x and $$y.

  3. What is the value of $$y when $$x=29?

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