Welcome to our direct variation calculator, where you can calculate the direct variation between two variables. If you're trying to determine the direct proportionality between two variables, you've come to the right place! In this article, we shall discuss the direct variation formula, real-life examples of direct variation, and how to find the constant of variation. We shall also graph the direct variation between two variables.
Direct variation of two variables
Direct proportionality or direct variation is the relationship between two variables such that an increase in one variable results in a proportional increase in the other variable. We express this relationship mathematically as:
y ∝ x
where:
- y — a dependent variable; and
- x — an independent variable.
By introducing a constant of proportionality k, we obtain the direct variation formula:
y = k · x
The graph of a direct variation will naturally be a straight line through the origin whose slope equals the constant k.
Examples of direct variation
Before learning how to find the constant of variation k, let's look at some examples of direct variation in different fields:
- Ohm's law: The current I flowing through a circuit directly varies with the circuit voltage V. The circuit resistance R is the constant of direct variation here:
V = I · R. - Cost and quantity: The total cost of an item varies directly with the number of units you buy. If one notebook costs $3, then
cost = 3 · quantity, and the price per notebook is the constant of variation. - Distance at constant speed: The distance travelled varies directly with time when speed is constant:
d = v · t, and the speed v is the constant of variation. - Circumference of a circle: The circumference varies directly with the diameter:
C = π · d, and here the constant of variation is the famous number π.
How do you find the constant of variation?
If you know one pair of matching values (x, y) that satisfy a direct variation, finding the constant of variation is easy. Start from the direct variation formula and solve for k:
y = k · x ⇒ k = y / x
For example, if y = 12 when x = 4, then k = 12 / 4 = 3, and the relationship is y = 3x. Once you know k, you can predict y for any x — or predict x for any y.
Using this direct variation calculator
Our tool works in four handy modes — just pick the one that matches what you know:
- Find the constant of variation k — enter a matching pair
(x, y)and we returnk = y / xtogether with the equationy = kx. - Find y — enter the constant k and a value of x; we compute
y = k · x. - Find x — enter the constant k and a value of y; we compute
x = y / k. - Missing value from a known pair — enter a known pair
(x₁, y₁)and a second valuex₂; we find the constant and theny₂ = k · x₂.
You can type decimals like 0.5, negative numbers like -3, or simple fractions like 3/4 in any field. An optional unit selector lets you label the values in metric units, US customary units, or world currencies (US dollar, Russian ruble, euro, and more) — it never changes the underlying math. The calculator also draws the line y = kx and shows a small table of values so you can see the proportional relationship at a glance.
Frequently Asked Questions
What is direct variation?
Direct variation is a relationship between two variables in which one is a constant multiple of the other: y = k · x. As x increases, y increases in the same proportion, and their ratio y / x always equals the constant of variation k.
How do you find the constant of variation?
Divide any matching output by its input: k = y / x. For instance, if y = 12 when x = 4, then k = 12 / 4 = 3. The constant is the same for every valid pair in a direct variation.
What does the graph of a direct variation look like?
It is always a straight line that passes through the origin (0, 0). The slope of that line equals the constant of variation k. A steeper line means a larger k, and a negative k makes the line fall from left to right.
What is the difference between direct and inverse variation?
In direct variation, y = k · x: both variables grow together. In inverse variation, y = k / x: as one variable increases, the other decreases. The direct variation calculator handles the first case.
Can the constant of variation be negative or a fraction?
Yes. The constant k can be any nonzero number — negative, a fraction, or a decimal. This calculator accepts values like -2, 3/4, or 0.25, and it can display the constant as a fraction when that is exact.