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Direct Variation
A relationship between two variables can be described by an equation or a formula. This relationship can be linear, quadratic, square root, or almost any other type of function you can think of. We will focus here on a linear relationship between two variables where one is a constant multiple of the other. This is a special relationship called direct variation.

In general, we say that y varies directly as x if there is a constant k so that the equation is true. When graphed, the constant k will be the slope of the line, y = mx + b.

In most cases, it will be your job to find the constant which makes the relationship true and then use that information to solve another problem.

To solve direct variation problems, you will need to be able to solve one-step equations, linear equations and sometimes literal equations. If you would like to get help and/or review solving one-step equations, click here; if your would like to get help and/or to review solving literal equations, click here; or if your would like to get help and/or to review the properties of linear equations, click here.

Let's Practice:
  1. Suppose y varies directly as x, and y = 20 when x = 4. Find y when x = 9.
When we see the phrase “y varied directly as x” that tells us when should be using the equation .

The next piece of information tells use that y = 20 and x = 4. We will use this information to find the value of k.
Now use the fact that k = 5 to find the value of y that the problem is asking for.
You should be careful to note that it is possible for other variables to be used in a problem. Based on the wording in the problem, this will tell you how to set up your initial equation.
  1. Suppose z varies directly as x, and z = 15 when x = 2. Find z when x = 8.
Note that there are different variables, but the initial equation used to find k will be similar to what was used before. Once we find k, then we use our equation again to find z.
Now use that value to find z.

Examples
Example Suppose m varies directly as p, and m = 22 when p = 8. Find p when m = 90.
What is your answer?
 
Example Suppose x varies directly as y, and x = 8 when y = 10. Find x when y = 16.
What is your answer?
 
Example Suppose v varies directly as g, and g = 70 when v = 30. Find v when g = 9.
What is your answer?
 
Example Suppose a varies directly as b, and a = 81 when b = 10. Find a when b = 100.
What is your answer?
 
Example Suppose y varies directly as x, and y = 9 when x = 3/2. Find y when x = 1.
What is your answer?
 



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