13.09 Linear rules
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 ($x$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?
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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.
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.
ANo, 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.
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.
AThe $y$y-value is equal to five times the $x$x-value.
BThe $y$y-value is equal to six times the $x$x-value.
CThe $y$y-value is equal to the $x$x-value.
DWrite 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?