Explore the Lesson

B. Linear Relations

A linear relation is a proportional relationship between the independent and dependent variables of a set of data. The graph of a linear relation is a straight line.

Linear Relation
a relation whose graph is a straight line

 

Example 1
Use the Cartesian coordinate plane to graph each of the following relations.
Determine whether the relations are linear.


a. {(−2,5), (−1,3), (0,1), (1,−1), (2,−3)}

This relation is linear. The plotted points in blue can be connected by a straight line.

b. {(2,4), (1,2), (0,0), (−1,−3), (−2,−4)}

This relation is not linear. The plotted points in red cannot all be joined by a straight line. The point (−1,−3) is not on the line that connects the other points.

 


When a relation is given as table of values or a set of ordered pairs, plotting the corresponding points is one way to determine whether the relation is linear. If the points can be connected by a straight line, the relation is linear.

Another way to determine whether a relation is linear is to identify whether there is a common difference between adjacent x-values and a common difference between adjacent y-values.

Example 2

Determine whether or not the following relation is linear by identifying whether there is a common difference between adjacent x-values and a common difference between adjacent y-values.

a.

x 5 6 7 8 9
y 10 8 6 4 2

x 6 − 5 = 1 7 − 6 = 1 8 − 7 = 1 9 − 8 = 1
y 8 − 10 = −2 6 − 8 = −2 4 − 6 = 2 2 − 4 = −2

Since the difference between adjacent x-values is common for the entire relation and the difference between adjacent y-values is common for the entire relation, the relation is linear. When x increases by 1, y decreases by 2.

b.

x 1 3 5 7 9
y 2 7 12 16 22

x 3 − 1 = 2 5 − 3 = 2 7 − 5 = 2 9 − 7 = 2
y 7 − 2 = 5 12 − 7 = 5 16 − 12 = 4 22 − 16 = 6

The difference between adjacent x-values is common for the entire relation, but the difference between adjacent y-values is not common for the entire relation, so the relation is not linear.