Lesson 69: Direct Variation

New Concept

Direct variation describes a relationship where two quantities increase or decrease together at a constant rate. When two variables are proportional, the value of one variable can be found by multiplying the other by a constant factor. We call this relationship between the variables direct variation.

In this equation, $x$ and $y$ are the variables, and $k$ is the constant multiplier, called the constant of proportionality.

What’s next

Now that you have the basics, you'll learn to identify direct variation from tables and graphs and solve problems using the constant of proportionality.

Property

When two variables are proportional, one value is found by multiplying the other by a constant factor, $k$. This relationship is defined by the equation:

Examples

Your pay depends on hours worked: $Total Pay = 12 \text{ dollars} \times hours$.
The perimeter of a square depends on its side length: $P = 4 \times s$.

Explanation

Think of it like this: whatever the independent variable (x) does, the dependent variable (y) copies it, but scaled by the special number k. If x doubles, y doubles. If x is cut in half, so is y. Their ratio is always constant and predictable.

Property

The constant of proportionality, $k$, is the fixed ratio in a direct variation, calculated as the dependent variable ($y$) divided by the independent variable ($x$).

Examples

If pay is 36 dollars for 3 hours, the constant is $k = \frac{36}{3} = 12$.
If Eunice says -2 and Doubleday replies -4 in the equation $d=ke$, then $k = \frac{-4}{-2} = 2$.

Explanation

This is the relationship's "magic number"! It’s the one value that connects the two variables and tells you their secret. To find it, just divide the output (y) by the input (x). It never, ever changes for that specific relationship.