### 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) = kx for some constant k.
(4)  y/x  is constant.
(5) the graph of = f(x) is a nonvertical line through the origin.

Example: In the equation y = 5x, 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 = 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 = 5x/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 = 8y, so that y15.

(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) 7x
B) x7
C) x + 7
D) 7/x

More Problems with Explanations

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