Direct Variation

Direct Variation

Today I would like to talk about **direct variation** – a topic that shows up on standardized tests such as the ACT, SAT, and GRE.

The following are all equivalent ways of saying the same thing:

(1) *y* varies directly as *x. *

(2) *y* is directly proportional to *x.*

(3) *y *= *kx* for some constant *k.*

(4) *y*/*x *is constant.

(5) the graph of *y *= *f*(*x*) is a nonvertical line through the origin.

**Example:** In the equation *y* = 5*x*, *y* varies directly as *x*. Here is a partial table of values for this equation.

Note that we can tell that this table represents a direct relationship between *x* and *y* because 5/1 = 10/2 = 15/3 = 20/4. Here the **constant of variation** is 5.

Here is a graph of the equation.

Note that we can tell that this graph represents a direct relationship between *x* and *y* because it is a nonvertical line through the origin. The constant of variation is the slope of the line, in this case *m *= 5.

The various equivalent definitions of direct variation lead to several different ways to solve problems.

**Example: **If *y* = *kx* and *y* = 5 when *x* = 8, then what is *y* when *x* = 24 ?

Try to solve the problem yourself before checking the solutions below.

**Solutions: **

**(1) **We are given that *y* = 5 when *x* = 8, so that 5 = *k*(8), or *k* = 5/8. Therefore *y* = 5*x*/8. When *x* = 24, we have *y* = 5(24)/8 = **15**.

**(2) **Since *y* varies directly as *x*, *y*/*x *is a constant. So, we get the following ratio: 5/8 = *y*/24. Cross multiplying gives 120 = 8*y*, so that *y* = **15**.

**(3) **The graph of *y* = *f*(*x*) is a line passing through the points (0,0) and (8,5) The slope of this line is (5 – 0)/(8 – 0) = 5/8. Writing the equation of the line in slope-intercept form we have *y* = 5/8 *x*. As in solution 1, when *x* = 24, we have *y* = 5(24)/8 = **15**.

**(4) **To get from *x* = 8 to *x* = 24 we multiply *x* by 3. So we have to also multiply *y* by 3. We get 3(5) = **15**.

Here are a few more problems for you to try. I will provide solutions to these over the next few days.

1. If *y* = *kx* and *y* = 7 when *x* = 11, then what is *y* when *x* = 33?

2. The amount of revenue that an online magazine retailer makes in a month is directly proportional to the number of active subscribers to the magazine. In July, the magazine had a total of 1200 subscribers, and the retailer reported revenue of $7200. In August, the online magazine had a total of 1500 subscribers. How much revenue did the retailer make?

3. If *y* is directly proportional to *x*, which of the following could express *y* in terms of *x* ?

A) 7*x*

B) *x*^{7}

C) *x* + 7

D) 7/*x*

**More Problems with Explanations**

If you are preparing for the SAT, ACT, or an SAT math subject test, you may want to take a look at the Get 800 collection of test prep books.

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