Direct variation occurs when as a variable increases, another variable increases by the same factor.  Direct variation is sometimes called direct proportion, since the variables are proportional to each other.  The opposite of direct variation is inverse variation.

We write:  y = kx

We say:  y varies directly as x

In the equation y = kx, k is called the constant of variation.

If we start with equation y = kx and divide both sides by x, we get y/x = k.  Therefore in direct variation, the quotient of the variables gives us the constant of variation. 

If you are presented with a table of (x,y) values and asked whether the table of values represent a direct variation, then it must by true that y/x = k for every value in the table.  Test each value by dividing y by x; if you get the same constant value, then you have a direct variation.

We can compare y = kx to the slope-intercept form of a line, y = mx + b.

We can see that y = kx is a linear equation with slope k and y-intercept 0. 

This tells us that the graph of a direct variation is a line that passes through the origin, point (0,0).  Therefore if you are asked whether a graph represents direct variation, the answer is "no" if it is not a straight line going through the origin. 

Examples of direct variation

Finding the constant of variation:

If y varies directly as x, and y = 4 when x = 2, then find the constant of variation and write an equation that expresses the direct variation.

Solution:  If y varies directly as x, then y = kx (definition of direct variation).  When y = 4 then x = 2, we plug these values into the equation y = kx.  We get 4 = k * 2.  Divide both sides by 2 and k = 2.  So our constant of variation is 2.  Therefore, we substitute 2 for k in our direct variation equation to get:  y = 2x.

Word Problem:

If distance traveled varies directly with the amount of gas used, and a car uses 7 gallons of gas to travel 280 miles, how much gas will be used to travel 450 miles?

Solution:  If mileage varies directly with gas used, then we can write m = kg.  We know that 7 gallons of gas was used to travel 280 miles, so we can substitute these values into the equation:  m = kg => 280 = k*7.  Solving, we get k = 40, so our equation becomes m = 40g.  I know I want my milage to be 450 miles, so I plug that into the equation and solve for g, the amount of gasoline:  450 = 40g; divide both sides by 40 and get g = 11.25 gallons.

These class notes provide further explanation and examples.