Modeling using Variation

There are three types of variations:

• Direct variation
• Inverse variation
• Joint variation

Direct Variation

Direct variation describes a simple relationship between two variables . We say y varies directly with x (or as x , in some textbooks) if:

y = kx

for some constant k , called the constant of variation or constant of proportionality . (Some textbooks describe direct variation by saying " y varies directly as x ", " y varies proportionally as x ", or " y is directly proportional to x .")

This means that as x increases, y increases and as x decreases, y decreases—and that the ratio between them always stays the same.

Inverse Variation

While direct variation describes a linear relationship between two variables , inverse variation describes another kind of relationship.

For two quantities with inverse variation, as one quantity increases, the other quantity decreases.

An inverse variation can be represented by the equation xy = k or $y={\frac {k}{x}}$ .

That is, y varies inversely as x if there is some nonzero constant k such that, xy = k or $y={k \over x}$ where x ≠ 0, y ≠ 0 .

Joint Variation

Joint variation occurs when a variable varies directly or inversely with multiple variables.

For instance, if x varies directly with both y and z, we have x = kyz. If x varies directly with y and inversely with z, we have $x={ky \over z}$ .

Notice that we only use one constant in a joint variation equation.

Resources

• Modeling Using Variation, Lumen Learning
• Intro to Direct & Inverse Variation, Khan Academy
• Direct Inverse and Joint Variation Word Problems, The Organic Chemistry Tutor