Identify proportional relationships

A proportional relationship between two quantities is one in which the two quantities vary directly with one and other. If one item is doubled, the other, related item is also doubled. Because of this, it is also called a direct variation.

The equations of such relationships are always in the form y = mx , and when graphed produce a line that passes through the origin.

In this equation, m is the slope of the line, and it is also called the unit rate, the rate of change, or the constant of proportionality of the function.

Consider a freight train moving at a constant speed of 30 miles per hour. The equation that expresses the distance travelled at that speed in terms of time is d=30t.

Is this a directly proportional relationship?

The graph of a proportional relationship is a straight line that passes through the origin.
Proportional quantities can be described by the equation y = kx, where k is a constant ratio.

You can tell that the relationship is directly proportional by looking at the graph. The graph is a straight line and it passes through the origin. So, the relationship is directly proportional.

You can also confirm that the linear relationship is directly proportional by showing that the relationship can be written as y = kx, where k is a constant ratio.

First, create a chart. Use points from the graph, such as (1, 30), (2, 60), and (3, 90).

Distance (miles) (y)	30	60	90
Time (hours) (x)	1	2	3

Now divide "Distance (miles) (y)" by "Time (hours) (x)" to find the ratio (m).

Distance (miles) (y)	30	60	90
Time (hours) (x)	1	2	3
Ratio (m)	30	30	30

The ratio is constant (m = 30), so the relationship can be described by the equation y = 30x. This equation means that the distance (miles) is always 30 times the time (hours)

Because the relationship can be written as y = 30x, the relationship is directly proportional.