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
• Variation Word Problems, Open Text Intermediate Algebra
• Intro to Direct & Inverse Variation, Khan Academy
• Direct Inverse and Joint Variation Word Problems, The Organic Chemistry Tutor