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Middle Years

10.01 Linear rules

Lesson

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.

DoWe 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
  1. Is the relationship linear?

    Yes, the relationship is linear.

    A

    No, the relationship is not linear.

    B

    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
  1. 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

    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
  2. 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
  1. What is the value of $y$y when $x=0$x=0?

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

  3. 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
  1. 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=

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

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

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