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 ($x$x) with corresponding values of the dependent variable ($y$y).

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

Worked Example

Does the following table of values represent a linear relationship?

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

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

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

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

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

Careful!

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

Practice question

Question 1

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

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

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: $y=mx+c$y=mx+c.

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

Practice question

Question 2

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

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

Which of the following options describes the relationship between $x$x and $y$y?
The $y$y-value is equal to the $x$x-value plus $five$ .

A
The $y$y-value is equal to $five$ times the $x$x-value.
B
The $y$y-value is equal to $six$ times the $x$x-value.
C
The $y$y-value is equal to the $x$x-value.
D
Write the linear equation that describes this relationship between $x$x and $y$y.

Question 3

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

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

What is the value of $y$y when $x=0$x=0?
Write the linear equation expressing the relationship between $x$x and $y$y.
What is the value of $y$y when $x=-16$x=−16?

Question 4

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

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

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

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

$m=\editable{}$m=

$c=\editable{}$c=
Write the linear equation expressing the relationship between $x$x and $y$y.
What is the value of $y$y when $x=29$x=29?

10.01 Linear rules

Identifying linear relationships

From a table of values

Worked Example

Practice question

Question 1

Writing rules for relationships

Practice question

Question 2

Question 3

Question 4

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