# Direct Variation Calculator

Created by Krishna Nelaturu
Reviewed by Anna Szczepanek, PhD and Adena Benn
Last updated: Jan 18, 2024

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:

$y \propto x$

Where:

• $y$ - A dependant variable; and
• $x$ - An independent variable.

By introducing a constant of proportionality $k$, we obtain the direct variation formula:

$y = k \cdot x$

The graph of a direct variation will naturally be a straight line 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:
$\qquad V = I \cdot R$

• Newton's second law: The rate of change of the velocity of a body is directly proportional to the force applied.
\qquad \begin{align*}\mathbf{F} &\propto \frac{\text{d}\mathbf{v}}{\text{dt}}\\[1em] \mathbf{F} &= m \frac{\text{d}\mathbf{v}}{\text{dt}} = m \mathbf{a} \end{align*}

Where:

• $\mathbf{F}$ - Force applied;
• $\mathbf{v}$ - Velocity of the body;
• $m$ -Mass of the body; and
• $a$ - Acceleration of the body.

Our Newton's second law calculator will assist you in force calculations and further reading on Newton's second law.

## How do you find the constant of variation?

To find the constant of direct variation k in the equation y = k ∙ x, follow these steps:

1. Measure the value of y for an arbitrary choice of x.
2. Divide this y by x to get k as k = y/x.
3. Verify your result using our direct variation calculator.

## Using this direct variation calculator

This direct variation equation calculator is straightforward to use:

1. Enter the value of the independent variable x.
2. Provide the proportionality constant k.
• The direct variation calculator will determine the value of the dependent variable y.
• The calculator will also provide a graph of the direct variation so you can visually assess the direct proportionality.

Thanks to the versatility of this calculator, you can enter any two known parameters to find the third.

Check out how inverse proportionality works using our inverse variation calculator.

## FAQ

### How do you recognize direct variation?

Follow these steps to identify a direct variation between two variables x and y:

1. Obtain multiple measurements of the variables, x1, y1, x2, y2, x3, y3,..., xn, yn.
• If the ratio of these variables is equal, y1/x1 = y2/x2 = y3/x3 = ... = yn/xn, the variables are directly varying.
• Alternatively, plot the readings as points on a graph, (x1, y1), (x2, y2), (x3, y3),..., (xn, yn). If they form a straight line, the variables are directly varying.

### What is y in the direct variation y = 12x, at x = 8?

y = 96.

1. Substitute x = 8 in the equation y = 12x to get y = 12 ∙ 8.
2. Compute the multiplication to get y = 96.
3. You can use our direct variation calculator for verification or further calculations.

### What is the relation between the mass of objects and gravitational force?

Newton's law of universal gravitation states that the gravitational force (F) between any two objects is directly proportional to the product of their masses (m1,m2). Mathematically, F ∝ m1∙m2.

Krishna Nelaturu
y = k ∙ x
Dependant variable (y)
Independent variable (x)
Direct proportionality constant (k)
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