Direct Variation Calculator

Find the constant of variation k, write the direct variation equation y = kx, and predict values using proportional relationships.

Choose What to Solve

Result

Direct Variation Equation
y = 3x
k = 3
Constant k3
Equationy = 3x
Known Point(4, 12)
Predicted Valuey(7) = 21

Step-by-Step Solution

y = kx => k = y/x = 12/4 = 3

What Is Direct Variation?

Direct variation describes a linear relationship between two variables where one variable is a constant multiple of the other. When y varies directly with x, we write y = kx, where k is called the constant of variation (or constant of proportionality). The graph of a direct variation is always a straight line passing through the origin (0, 0).

In everyday language, direct variation means that as one quantity increases, the other increases proportionally, and as one decreases, the other decreases proportionally. The ratio y/x always remains constant and equal to k.

Key Concepts

Direct Variation Formula

The fundamental equation relating y and x through a constant k.

y = kx, where k = y/x

Finding k

Given any point (x, y) on the line, divide y by x to find the constant.

k = y / x (x cannot be 0)

Proportional Relationship

If (x1, y1) and (x2, y2) are both on the line, then y1/x1 = y2/x2.

y1/x1 = y2/x2 = k

How to Identify Direct Variation

A relationship between x and y is a direct variation if and only if:

  • The equation can be written in the form y = kx (no added constant).
  • The graph is a straight line that passes through the origin.
  • The ratio y/x is the same for all data points.
  • When x = 0, then y = 0.

Examples of Direct Variation

  • Distance and time at constant speed: d = rt (distance varies directly with time when rate is constant).
  • Circumference and diameter: C = pi * d (circumference varies directly with diameter, k = pi).
  • Hooke's Law: F = kx (spring force varies directly with displacement).
  • Wages: Pay = rate * hours (total pay varies directly with hours worked).
  • Unit pricing: Total cost = price per unit * quantity.

Direct Variation vs. Other Relationships

Direct variation is often confused with other types of mathematical relationships. Here are the key differences:

  • Direct variation (y = kx): Linear, passes through origin, constant ratio y/x.
  • Inverse variation (y = k/x): Hyperbolic curve, constant product x*y = k.
  • Joint variation (z = kxy): z varies directly with both x and y.
  • Linear (non-proportional) (y = mx + b, b not equal to 0): Line that does NOT pass through origin.

The Constant of Variation

The constant k tells you how steep the line is and in which direction it goes. If k > 0, both variables increase together (positive slope). If k < 0, as one increases the other decreases (negative slope). The absolute value of k indicates the rate of change: for every 1-unit increase in x, y changes by k units.

Graphing Direct Variation

To graph y = kx, simply plot the origin (0, 0) and use the known point (x, y) to establish the slope k. Draw a straight line through both points extending in both directions. The slope of this line is exactly k, making it easy to read off predicted values visually.