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.
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.
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?
In a linear relationship, the y-value must change by an equal amount each time the x-value changes by 1.
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.
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.
What is the value of y when x=29?
All linear relationships can be expressed in the form: