a linear equation can be used to relate the two variables
the graph of the relationship on a number plane is a straight line
the dependent variable changes by a constant amount as the independent variable changes

From a graph

When shown a graph that represents a relationship between two variables, if the graph is a straight line, then the graph is a linear relation and a linear equation can be used to represent the relationship. Below is an example of a linear relation:

In this chapter, linear graphs will be created using technology, such as a CAS calculator. Learning how to sketch linear graphs will be covered in more detail in Chapter 9.

From an equation

An equation is linear when the equation is or can be arranged into one of the following forms:

Linear equation forms

$y=a+bx$y=a+bx, where $y$y (the dependent variable) is the subject of the equation
$Ax+By=C$Ax+By=C where $A,B,C$A,B,C are constants

Careful!

The $x$x term in a linear relationship will always have a power of $1$1, though the power is rarely written explicitly.

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 such as the one above can be used to recognise a linear relationship, if there is a common difference between $y$y values as $x$x changes by a constant amount.

If the table has consecutive $x$x values: for each $1$1 unit change in $x$x, if the $y$y value changes by the same amount each time then the relationship between $x$x and $y$y must be a linear relation (i.e. the gradient is always the same).

Worked example

Example 1

Consider the table of values given below. Does the table represent a linear relation?

$x$`x`	$3$3	$4$4	$5$5	$6$6
$y$`y`	$12$12	$15$15	$18$18	$21$21

Some things to note about this table: the $x$x values go up by $1$1 each time and it doesn't matter what number the table starts at.

Notice in the above table that, for each $1$1 unit increase in $x$x, $y$y increases by $3$3. This is a linear relationship, as this constant change in $y$y indicates a common difference.

So we can check for a linear relationship by looking for a common difference between the $y$y values. If each successive $y$y value has the same difference then it is linear.

Practice questions

Question 1

$y$y is equal to $11$11 groups of $x$x more than $-1$−1.

Write the statement above as a mathematical equation.
Is this equation linear?
Yes
A
No
B

Question 2

Would the following table of values represent a linear graph?

$x$x $3$3 $6$6 $9$9 $12$12 $15$15

$y$y $-7$−7 $-14$−14 $-21$−21 $-28$−28 $-35$−35

Yes
A
No
B
$x$x $7$7 $21$21 $35$35 $49$49 $63$63

$y$y $\frac{15}{2}$152 $15$15 $\frac{45}{2}$452 $45$45 $\frac{135}{2}$1352

Yes
A
No
B

$x$`x`	$3$3	$6$6	$9$9	$12$12	$15$15
$y$`y`	$-7$−7	$-14$−14	$-21$−21	$-28$−28	$-35$−35

$x$`x`	$7$7	$21$21	$35$35	$49$49	$63$63
$y$`y`	$\frac{15}{2}$152	$15$15	$\frac{45}{2}$452	$45$45	$\frac{135}{2}$1352

Question 3

Consider the graph shown below.

Loading Graph...

Does this represent a linear equation?
Yes
A
No
B

Outcomes

U1.AoS4.1

the properties of linear functions and their graphs

2.01 Linear relations

Recognising linear relationships