 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 $y$y $1$1 $2$2 $3$3 $4$4 $5$5 $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 $y$y $1$1 $2$2 $3$3 $4$4 $5$5 $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 $y$y $1$1 $2$2 $3$3 $4$4 $5$5 $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

### 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 $y$y $1$1 $2$2 $3$3 $4$4 $5$5 $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
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 $y$y $1$1 $2$2 $3$3 $4$4 $5$5 $-2$−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 $y$y $0$0 $1$1 $2$2 $3$3 $4$4 $5$5 $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?