# 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:

#### 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.

#### 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).

#### 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:

## 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.