Linear Relations: Direct vs. Partial Variation

Linear relations can be difficult to learn, but once you get a grasp it should start to come very easily.  I’ve been an in-school math tutor for two years, and I frequently get asked the question, “What’s the difference between direct variation and partial variation?”  Understanding the two basic types of variations helps you to analyze and develop graphs, tables, and (slope – y-intercept) equations.

What is direct variation?  This is when a line on a graph passes through the origin (0, 0).  In other words, the y-intercept is 0. In a table of “x” and “y” coordinates, you can identify direct variation when you see the number “0” in both “x” and “y” columns.  In a slope – y-intercept equation, y=mx+b, there would be no “b.” This is because “b,” the y-intercept, is 0 when the relation is a direct variation.

What is partial variation?  This is when a line on a graph passes through anywhere on the y-axis except the origin.  The y-intercept could be (0, 3) or (0, 4), but NOT (0, 0). In a table showing coordinates of a linear relation, you can identify whether or not the relation is partial variation by find the “0” in the “x” column and seeing if the number in the ‘y” column is not “0.”  In a slope – y-intercept equation, the “b” (y-intercept) would be anything but 0.

To summarize, direct variation means the y-intercept is (0, 0) and partial variation means the y-intercept is anything but (0, 0)!


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