Problem 73 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The equation represents a direct variation with constant \( c \).

Step 1: Identify Equation Type

The given equation is \( \frac{a}{b} = c \). This equation can be rewritten as \( a = c \cdot b \) by multiplying both sides by \( b \). This shows that \( a \) varies directly with \( b \).

Step 2: Determine the Constant of Variation

In the equation \( a = c \cdot b \), the constant \( c \) is the number that relates \( a \) and \( b \). This constant is called the constant of variation. Since \( a \) is equal to \( c \) times \( b \), \( c \) is the constant of direct variation between \( a \) and \( b \).

Key Concepts

Constant of Variation

In mathematics, the constant of variation is a key component when dealing with relationships such as direct or inverse variation. When we talk about "variation," we refer to how one variable changes in relation to another. This concept is simplified through the constant of variation, which helps define the exact relationship between variables.
For instance, in a direct variation example represented as \( y = kx \), "\( k \)" is the constant of variation. This constant "\( k \)" remains unchanged in direct relationships and tells us how much the dependent variable (\( y \)) changes when the independent variable (\( x \)) changes.
This relationship can be visualized as a line passing through the origin, indicating a proportional increase or decrease between the two variables. Understanding and identifying the constant of variation contains the key to unlocking many real-world applications, such as calculating speed, density, or rate. Recognizing that the constant frames the relationship allows us to make comparisons and predictions across varying scenarios.

• Direct variation form: \( y = kx \)
• "\( k \)" stays constant, while \( y \) and \( x \) vary.
• Proportionate relationship: If \( x \) doubles, \( y \) doubles.

Inverse Variation

Inverse variation describes a relationship where one variable increases while the other decreases such that their product remains constant. This type of variation is represented by the equation \( xy = k \), where "\( k \)" is again the constant, but in this case, it signifies an inverse relationship.
To help students visualize this concept, think of two variables pulling on opposite ends of a rope. As one side pulls harder and moves further away, the other side moves closer in response to balance the tension on the rope. Here, the product of the two movements (their constant pull) remains unchanged.
In more practical terms, a common example of inverse variation is in the context of physics or mathematics problems involving speed and time. For example, if a car travels a certain distance, increasing speed will decrease travel time, and decreasing speed will increase travel time, as long as the distance remains unchanged.

• In the equation \( xy = k \), "\( k \)" remains constant.
• As one variable (\( x \)) increases, \( y \) must decrease to keep \( k \) the same.
• Reflection in graphs: Forms a hyperbola rather than a straight line.

Joint Variation

Joint variation occurs when a variable varies directly as the product of two or more other variables. This is expressed in the form \( z = kxy \), where "\( k \)" remains the constant of variation, and \( z \), \( x \), and \( y \) are the variables in play.
A simple way to understand joint variation is through its expression of a multi-faceted dependency. For example, the volume of a cylinder can be expressed by the relationship \( V = \pi r^2h \), where the volume \( V \) varies jointly with the square of the radius \( r \) and the height \( h \). The constant \( \pi \) signifies the proportional relationship governing the volume's dependence on the radius and height.
Joint variation allows us to analyze systems with multiple influencing factors easily. It emphasizes the interconnectivity of variables working in unison to affect a singular outcome.

• Equation form: \( z = kxy \)
• "\( k \)" ties multiple factors, highlighting a compound relationship.
• Examples involve multiple dimensions, like density = mass/volume.