Problem 16 - Algebra 2

Question

State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. \(\frac{n}{m}=1.5\)

Step-by-Step Solution

Answer

The equation represents a direct variation, and the constant of variation is 1.5.

Step 1: Identify the form of the given equation

We have the equation \( \frac{n}{m} = 1.5 \). This can be rewritten as \( n = 1.5m \).

Step 2: Determine the type of variation

The equation \( n = 1.5m \) is in the form of \( y = kx \), which represents a direct variation because one variable is directly proportional to the other.

Step 3: Identify the constant of variation

In the equation \( n = 1.5m \), the constant of variation, \( k \), is the coefficient of \( m \), which is 1.5.

Key Concepts

Constant of Variation

The constant of variation is a crucial concept in understanding different types of variable relationships in mathematics. It is represented by the symbol \( k \). In direct variation equations, which take the form \( y = kx \), the constant \( k \) signifies the rate at which \( y \) changes with respect to \( x \).
In our example, \( \frac{n}{m} = 1.5 \) translates into \( n = 1.5m \). Here, the constant of variation is \( 1.5 \). This means that for every unit increase in \( m \), \( n \) increases by 1.5 units.
Constant of variation provides a fixed multiplier that helps define the proportionality between variables:

• In direct variation, if one variable doubles, the other also doubles, provided \( k \) remains positive.
• It ensures predictability and uniformity in the relationship between the variables.

This constant can vary depending on the equation and the real-world situation it represents.

Inverse Variation

Inverse variation describes a scenario where one variable increases as the other decreases. In mathematical terms, this relationship is expressed as \( y = \frac{k}{x} \), where \( k \) is the constant of variation. Here, unlike direct variation, \( y \) and \( x \) are inversely related.
It's important not to confuse inverse variation with direct variation. While in direct variation, both variables move in the same direction, in inverse variation:

• When \( x \) increases, \( y \) decreases.
• When \( x \) decreases, \( y \) increases.

For example, if an equation involved \( y = \frac{5}{x} \), as \( x \) becomes larger, \( y \) becomes smaller and vice versa. The constant of variation \( k \) dictates how strongly \( y \) changes in response to changes in \( x \).
Inverse variation is often found in real-life scenarios such as the inverse relationship between speed and travel time when distance is constant.

Joint Variation

Joint variation combines direct variation with more than one variable. When a variable is directly proportional to the product of two or more other variables, it is considered to involve joint variation. A common form this takes is \( z = kxy \), where \( z \) varies jointly with \( x \) and \( y \) and \( k \) is the constant of variation.
In joint variation:

• Both \( x \) and \( y \) affect \( z \), meaning that changes in either one will influence \( z \).
• All variables increase or decrease simultaneously based on the same constant factor \( k \).
• This concept is applicable to scenarios like calculating the volume of a prism, where the volume is jointly proportional to the area of the base and the height.

Joint variation allows for more complex relationships and models how multiple factors can work together to affect an outcome in a unified manner.