Technical Mathematics, Sixth Edition

17.3. Direct Variation

▪ Exploration:

Try this. In the same viewing window, graph

y = x y = 2x y = 3x

(a) What happens to y, for each function, as x increases? (b) What is the effect of the coefficient of x?

If two variables are related, as in our exploration, by an equation of the form

y = kx

where k is a constant, we say that

y varies directly as x

or that

y is directly proportional to x

The constant k is called the constant of proportionality. Direct variation can also be indicated by using the special symbol ∞, which means is proportional to. With either symbol, we have what is called direct variation.

NOTE

17.3.1. Solving Variation Problems

Variation problems can be solved with or without evaluating the constant of proportionality. We first show a solution in which the constant is found by substituting the given values into Eq. 49.

TEACHING TIP: List the steps for this procedure: Substitute the known pair of x and y values into the formula, and solve for k. Rewrite the formula with the value for k. Substitute the single x value, and solve for y.

Example 19: If y is directly proportional to x, and y is 27 when x is 3, find y when x is 6. Solution: Since y varies directly as x, we use Eq. 49. y = kx To find the constant of proportionality, we substitute the given values for x and y, 3 and 27. 27 = k(3) So k = 9. Our equation is then y = 9x When x = 6, y = 9(6) = 54

We now ...