Problem 74 - Algebra 2

Question

State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. \(y=-7 x\)

Step-by-Step Solution

Answer

Direct variation; constant of variation is \(-7\).

Step 1: Identify the Equation Type

The equation given is \(y = -7x\). Notice that the variable \(y\) is equal to \(x\) multiplied by a constant \(-7\). This format indicates that the relationship between \(y\) and \(x\) is a direct variation, where \(y\) varies directly as \(x\).

Step 2: Determine the Variation Constant

In a direct variation, the equation has the form \(y = kx\), where \(k\) is the constant of variation. By comparing \(y = -7x\) with \(y = kx\), we identify that the constant of variation \(k\) is \(-7\).

Key Concepts

Constant of Variation

In the context of direct variation, understanding the constant of variation is crucial. The constant of variation, often denoted by the letter "k," is the fixed number that relates two variables in a direct or inverse proportionality.
The equation of direct variation is written as:

• Direct Variation: \( y = kx \)
• Inverse Variation: \( y = \frac{k}{x} \)

In a direct variation, the constant \( k \) signifies how much \( y \) changes for every unit increase in \( x \). For example, in the equation \( y = -7x \), the constant of variation \( k \) is \(-7\).
This means that for every unit increase in \( x \), \( y \) decreases by 7 units. Similarly, but inversely, an inverse variation involves a relation where as one quantity increases, the other decreases, maintaining the product constant. Hence, understanding the constant helps explain how two variables depend directly or inversely on each other.

Inverse Variation

Inverse variation describes a relationship where the product of two variables is constant. As one variable increases, the other decreases to maintain the product constant. The mathematical expression for inverse variation is given by:

• \( y = \frac{k}{x} \)

Here, \( k \) is the constant of variation, and it indicates the invariant product of \( y \) and \( x \).
For example, if \( y \) varies inversely as \( x \) and the constant \( k \) is 12, the equation is \( y = \frac{12}{x} \). As \( x \) increases, \( y \) decreases, but their product \( y \cdot x = 12 \) remains steady.
This type of variation is especially relevant in real-world scenarios where inverse relationships are common, such as speed and time during travel.

Joint Variation

Joint variation occurs when a single quantity is proportional to the product of two or more other quantities. When a variable varies jointly as multiple variables, it means that it is directly proportional to the product of those variables. The formula for joint variation is:

• \( z = kxy \)

In this case, \( z \) varies jointly as \( x \) and \( y \), and \( k \) is the constant of variation.
Suppose \( z \) varies jointly with \( x \) and \( y \), where \( k \) is 5, the equation would be \( z = 5xy \).
This implies that any change in \( x \) or \( y \) will proportionally affect \( z \), multiplied by the constant \( k \). Joint variation is useful in many fields, including physics, where multiple factors influence a given outcome simultaneously. It helps to model complex relationships where multiple variables are interdependent.