What is 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 a gallon of milk costs $3, and I buy 1 gallon, the total cost is$3. If I buy 10 gallons, the price is \$30. In this example the total cost of milk and the number of gallons purchased are subject to direct variation -- the ratio of the cost to the number of gallons is always 3.

To be more "geometrical" about it, if y varies directly as x, then 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 variation. That's because each of the variables is a constant multiple of the other, like in the graph shown below:

Key concepts of direct variation:

1) How do I Recognize Direct Variation in an Equation?

The equation $$\frac{y}{x} = 6$$ 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 $$\frac{y}{x} = 6$$ is called the constant of variation. The equation $$\frac{y}{x} = 6$$ can also be written in the equivalent form, $$y = 6x$$. That form shows you that y is always 6 times as much as x.

Similarly, for the equation $$y=\frac{x}{3}$$, the constant of variation is $$\frac{1}{3}$$. The equation tells us that for any x value, y will always be 1/3 as much.

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. If we double x, then we also double the corresponding y value. 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. That equation tells us that the perimeter is always four times the length of a single side (makes sense, right?), but it also tells us that doubling the length of the sides doubles the perimeter (which will still be four times larger in total).

3) Geometric Interpretation of Direct Variation

The equation $$y = kx$$ is a special case of linear equation ($$y=mx+b$$) where the y-intercept equals 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 straight 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 $$y = mx$$, m is the constant of variation.

Example A:

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

Plan of Attack:

Plug the given values into the equation $$y = kx$$.

Solve for k.

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

Step-by-Step:

Start with our standard equation: $$y = kx$$

Insert our known values: $$8 = k*12$$

Divide both sides by 12 to find k: $$\frac{8}{12} = k$$

$$\frac{2}{3} = k$$

Next: Go back to $$y = kx$$ and replace k with $$\frac{2}{3}$$.

Result:

$$y = \frac{2}{3}*x$$

Example B:

If y varies directly as x, and $$y = 24$$ when $$x = 16$$, find y when $$x = 12$$.

Plan of Attack:

When two quantities vary directly, their ratio is always the same. We'll create two ratios, set them equal to each other, and then solve for the missing quantity.

Step-by-Step:

The given numbers form one ratio which we can write as $$\frac{y}{x}$$: $$\frac{24}{16}$$

To find y when $$x=12$$ we setup another ratio: $$\frac{y}{12}$$

Solve:

By definition, both ratios are equal:

$$\frac{24}{16} = \frac{y}{12}$$

Multiply each side by 12 to solve for y:

$$\frac{24}{16}*12 = y$$ $$y = \frac{3}{2}*12$$

Result:

y = 18 when x = 12

Got a basic understanding of direct variation now? If you still need more help, try searching our website (at the top of the page) for a more specific question, or browse our other algebra lessons. Sometimes it helps to have a subject explained by somebody else (a fresh perspective!) so you may also be interested in another lesson on direct variation, such as this page that provides examples solving direct variation.