Understanding Partial Variation

What is Partial Variation?

Partial variation occurs when a variable depends on two or more other variables, where one part of the relationship is direct and the other is constant. It is a combination of direct variation and a constant.

For example:

• If a quantity y varies directly as x and has a constant part c, it can be expressed as: $y = kx + c$, where $k$ is the constant of variation.

Characteristics of Partial Variation

• Combination of Variations: It includes both a direct variation component and a constant component.
• Linear Relationship: The graph of a partial variation is a straight line.
• Constant Shift: The constant part shifts the line up or down on the graph.

How to Identify Partial Variation

1. Identify the Variables: Determine which variable is dependent and which is independent.
2. Look for Direct Variation: Check if part of the relationship is directly proportional.
3. Find the Constant: Identify the constant part that does not change with the independent variable.

Examples

Example 1: Understanding Partial Variation

Problem: A taxi fare is calculated as $\text{Fare} = 50x + 200$ where $x$ is the distance traveled in kilometers. Explain the components of this equation.

Example 2: Formulating Partial Variation

Problem: A worker's weekly pay $P$ is given by $P = 20h + 100$, where $h$ is the number of hours worked. Identify the elements of partial variation.