Problem 72 - Algebra 2

Question

State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. \(m n=4\)

Step-by-Step Solution

Answer

The equation represents an inverse variation with a constant of variation of 4.

Step 1: Identify the Type of Variation

The given equation is \( mn = 4 \). To identify the type of variation, we need to rearrange the equation to see if it fits a standard form. In this case, rearranging gives \( n = \frac{4}{m} \). This form indicates that \( n \) varies inversely with \( m \), because an inverse variation occurs when one variable is equal to a constant divided by the other variable.

Step 2: Determine the Constant of Variation

In the equation \( mn = 4 \), the constant of variation, commonly denoted by \( k \), is the product of the two variables \( mn \). Here, \( k = 4 \), which is the constant provided in the equation. Thus, 4 is the constant of variation indicating how \( m \) and \( n \) inversely vary with respect to each other.

Key Concepts

Direct Variation

Direct variation describes a relationship between two variables where one is a constant multiple of the other. The general form of a direct variation is given as \( y = kx \), where \( y \) and \( x \) are the variables and \( k \) is the constant of variation. This kind of variation implies that if you increase one variable, the other increases proportionally.
For example, if \( k \) is 2, for each unit increase in \( x \), \( y \) will increase by 2 units.

• This relationship is linear and directly proportional.
• It can be represented on a graph as a straight line passing through the origin.

Direct variation is different from inverse variation, where one variable increases as the other decreases, like in the equation \( mn = 4 \).
Remember, in direct variation, both variables move in tandem.

Joint Variation

Joint variation involves two or more variables that vary directly. This means that a variable can be dependent on the product of two or more other variables. The standard form can be written as \( z = kxy \), where \( z \) varies jointly with \( x \) and \( y \), and \( k \) is the constant of variation.

• An increase in either \( x \) or \( y \) will result in an increase in \( z \).
• It is an extension of direct variation, with more variables involved.

For instance, if \( k = 3 \), \( x = 2 \), and \( y = 4 \), then \( z = 3 \times 2 \times 4 = 24 \).
This kind of relationship is common in formulas available in physics and chemistry, where one quantity is dependent on multiple others simultaneously.

Constant of Variation

The constant of variation is a non-zero number that relates how the variables in a variation problem affect each other. In direct variation, \( y = kx \), it defines the rate at which \( y \) changes with \( x \). For inverse variation, like the equation \( mn = 4 \), it is the constant product of the variables that equal \( k \).

• It functions as a measure of proportionality in direct variation.
• In inverse variation, it establishes the fixed rate of multiplication of two inversely dependent variables.

Knowing the constant of variation helps in solving real-world problems, making predictions based on variable changes. For example, in the exercise given, the constant of variation allows one to see how if one variable grows, the other must shrink accordingly to maintain the constant product of 4.
Understanding and identifying \( k \) is crucial in distinguishing between different types of variations.