Direct Variation

The math says that ­y varies with x.  What does this mean?

Direct variation is a relationship between two variables, x and y.  When two variables vary directly, their ratio (the fraction ) always equals the same number.  That number is called the Constant of Variation.  When we don’t know the value of the constant of variation, we use the letter k to hold its place. 

The values of x and y change based on the value of k, but their values also depend heavily on each other.  In direct variation, whenever x gets bigger, y gets bigger.  Whenever x gets smaller, y gets smaller.  The reverse is also true.  Whenever y gets bigger, x gets bigger.  Whenever y gets smaller, x gets smaller. 

The Direct Variation Equation

The general equation for direct variation is written

y = kx

We can read this in two ways: “y varies directly as x,” or “y equals k times x.”

Whenever we need to find the value of k, we can divide both sides of the equation by x.  The resulting equation looks like this:

How do I tell if an equation has direct variation?

A direct variation equation can only have multiplication or division in it.  A direct variation equation will never involve addition or subtraction.

If you are given a set of related x and y values, plug each pair into the equation

If you get the same value for k every time, then you have direct variation.

If you are looking at a graph, direct variation has two defining characteristics.  The graph will be a straight line, and that line will go through the origin, which is the point (0, 0).

How do I solve a problem with direct variation?

Direct variation problems will often involve two major steps.  The math will usually give you either a related pair of values for x and y, or they will give you an ordered pair (our notation for a point in the Cartesian Coordinate Plane; this is actually still a related pair of values).  The math will also give you a value for one of the variables, but not the other.  Your task is to find the missing related value.

Once you have identified your related values for x and y, your first step is to find k (the constant of variation).  Once you know the value of k, your second step is to solve for the missing value.

For Example

y varies directly with x, and y = 35 when x = 7.  What is the value of y when x = 9?

·         Prep Step: Identify your related values.

o   The problem tells us, “y = 35 when x = 7.”  These are our related values.

·         Step 1: Find the value of k.

o   We can plug our related values and k into the general equation for direct variation.  When we isolate k, we find its value.

y = kx

35 = k(7)

o   We can divide both sides of the equation by 7 to isolate k.

5 = k

·         Step 2: Find the missing value.

o   The problem asks, “What is the value of y when x = 9?”  We need to plug 9 in for x and 5 in for k, then solve for y.

y = 5(9)

y = 45

Another Example

Do these related sets of values for x and y represent direct variation?  If so, write an equation for the direct variation.

Set 1: (-2, 3), (-6, 7), and (4, -5)

Set 2: (4, 5), (8, 10), and (10, 12.5)

·         Step 1: Plug all of the given values into the second direct variation equation and solve for k.


·         Step 2: Compare the values of k that you found.

o   If the fractions do not reduce easily, you can use division to convert them to decimals.  To do so, you divide the numerator (top number) by the denominator (bottom number).

Set 1:

-1.5 = k, -1.167 = k, -1.25 = k

o   These values of k are all different.  This means that we do not have direct variation.

Set 2:

1.25 = k, 1.25 = k, 1.25 = k

o   These values of k are all the same.  This means that we do have direct variation.