Direct Variation

When two variables are related in such a way that the ratio of their values always remains the same, the two variables are said to be in direct variation.

In simpler terms, that means if A is always twice as much as B, then they directly vary.

If y varies directly as x, the graph of all points that describe this relationship is a line going through the origin (0, 0) whose slope is called the constant of the variation. That's because each of the variables is a constant multiple of the other.

Let's review several areas of direct variation.

1) Expressing Direct Variation an an Equation

The equation y/x = 6 states that y "varies directly as" x since the ratio of y to x (also written y : x ) never changes. The number 6 in the equation y/x = 6 is called the constant of variation. The equation y/x = 6 can also be written in the equivalent form,
y = 6x. That form shows you that y is always 6 times as much as x.

How did I come up with y = 6x? Basically, multiply BOTH sides of the equation by x to isolate y in terms of x. Think of y as a variable that loves to be alone. The general form of our sample equation y = 6x is written y = kx, where k is the constant of variation. In other words, the value of k does not change.

2) Algebraic Interpretation of Direct Variation

For an equation of the form y = kx, multiplying x by some fixed amount also multiplies y by the SAME FIXED AMOUNT. What does this mean? For example, since the perimeter P of a square varies directly as the length of one side of a square, we can say that P = 4s, where the number 4 represents the four sides of a square and s represents the length of one side.

3) Geometric Interpretation of Direct Variation

The equation y = kx is a special case of linear equation
y = mx + b, where b = 0. (Note: the equation y = mx + b is the slope-intercept form where m is the slope and b is the y-intercept). Anyway, a line through the origin (0,0) always represents a direct variation between y and x. The slope of this line is the constant of variation. In other words, in the equation
y = mx + b, m is the constant of variation.

Sample A:

If y varies directly as x and y = 8 when x = 12, find k and write an equation that expresses this variation.

Steps:

1) Plug the given values into the equation y = kx.

2) Solve for k.

3) Then replace k with its value in the equation y = kx.

y = kx

8 = k12

Divide both side by 12 to find k.

8/12 = k

2/3 = k

Next: Go back to y = kx and replace k with 2/3.

y = (2/3)x

Sample B: If y varies directly as x and y = 24 when x = 16, find y when x = 12.

Solution: Set up a proportion since the ratios of corresponding values of x to y are always the same.

(24/16) = (y/12)

Cross-multiply to find y.

24 times 12 = 16 times y

288 = 16y

Next: Divide both sides by 16 to find the value of y.

18 = y

By Mr. Feliz
(c) 2005