A relationship between two variables is linear if both of the following conditions are met:
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.
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.
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 $y$yvalue must change by equal amounts as the $x$xvalue increases by $1$1. We can see that the $x$xvalues in this table of values are increasing by $1$1 each step, so we want to find out if the $y$yvalues 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 $y$yvalue at each step. We can see straight away that the $y$yvalue is increasing for each step, but by how much?


We can see that the $y$yvalue always increases by $8$8 as the $x$xvalue increases by $1$1. This means the table of values does represent a linear relationship.
Reflect: By comparing the increases in the $y$yvalue as the $x$xvalue increases by $1$1, we can determine if an equation is linear.
The $x$xvalues 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 $y$yvalue by the increase in the $x$xvalue to find the unit change.
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.
No, the relationship is not linear.
Yes, the relationship is linear.
No, the relationship is not linear.
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.
All linear relationships can be expressed in the form: $y=mx+c$y=mx+c.
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 $y$yvalue is equal to the $x$xvalue plus five.
The $y$yvalue is equal to five times the $x$xvalue.
The $y$yvalue is equal to six times the $x$xvalue.
The $y$yvalue is equal to the $x$xvalue.
The $y$yvalue is equal to the $x$xvalue plus five.
The $y$yvalue is equal to five times the $x$xvalue.
The $y$yvalue is equal to six times the $x$xvalue.
The $y$yvalue is equal to the $x$xvalue.
Write the linear equation that describes this relationship between $x$x and $y$y.
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?
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?