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.

Ideas

Table of values
Rules for relationships

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) with corresponding values of the dependent variable (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 values as x changes by a constant amount, then there is a linear relationship

The x-values in a table of values might not necessarily increase by 1 each step. However, we can still use this method by dividing the increase or decrease in the y-value by the increase in the x-value to find the unit change.

Examples

Example 1

Consider the relationship between x and y in the table below.

x	1	2	3	4	5
y	5	1	-3	-7	-11

Is the relationship linear?

Worked Solution

Create a strategy

In a linear relationship, the y-value must change by equal amounts as the x-value changes by 1.

Apply the idea

We can see that the x-values are increasing by 1 each time.

For the y-values: the change from 5 to 1 is 1-4=-4. From 1 to -3 is -3-1=-4.

We can see that the y-values continue to decrease by 4 each time.

Yes, the relationsip is linear.

Idea summary

In a linear relationship, the y-value must change by an equal amount each time the x-value changes by 1.

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.

m is equal to the change in the y-values for every increase in the x-value by 1.
c is the value of y when x=0.

Examples

Example 2

The variables x and y are related, and a table of values is given below:

x	0	1	2	3	4	5
y	8	13	18	23	28	33

Linear relations can be written in the form y=mx+c. Find the linear equation for this relationship.

Worked Solution

Create a strategy

Remember, m is equal to the change in the y-values for every increase in the x-value by 1, and c is the value of y when x=0.

Apply the idea

As the x-values increase by 1 we can see in the table that the y-values increase by 5: \\ 8+5=12, \, 13+5=18, \, 18+5 = 23, \ldots

So m=5.

The value of y when x=0 is 8, so c=8.

The linear equation is: y=5x+8

What is the value of y when x=29?

Worked Solution

Create a strategy

Substitute the given x-value into the equation.

Apply the idea

\displaystyle y	\displaystyle =	\displaystyle 5x+8	Write the equation
\displaystyle y	\displaystyle =	\displaystyle 5 \times 29 + 8	Substitute x=29
	\displaystyle =	\displaystyle 145+8	Evaluate the multiplication
	\displaystyle =	\displaystyle 153	Evaluate

Idea summary

All linear relationships can be expressed in the form:

\displaystyle y=mx+c

\bm{m}

is the change in the y-values for every increase in the x-value by 1

\bm{c}

is the value of y when x=0

Outcomes

MA4-11NA

creates and displays number patterns; graphs and analyses linear relationships; and performs transformations on the Cartesian plane

6.01 Linear rules

Introduction

Ideas

Table of values

Examples

Example 1

Rules for relationships

Examples

Example 2

Outcomes

MA4-11NA

What is Mathspace

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