Problem 16 - Elementary and Intermediate Algebra

Question

Complete each solution. Determine whether each equation defines inverse variation. a. \(y=k x\) b. \(y=\frac{k}{x}\) c. \(y=\frac{x}{k}\) d. \(d=\frac{k}{g}\)

Step-by-Step Solution

Answer

Equations b and d define inverse variation.

Step 1: Understand Inverse Variation

Inverse variation means that two variables are related in such a way that their product is constant. This relationship can be expressed as \( y = \frac{k}{x} \), where \( k \) is a constant.

Step 2: Analyze Equation a

The equation given is \( y = kx \). This represents direct variation since it is in the form \( y = kx \) where \( k \) is a constant, not inverse variation.

Step 3: Analyze Equation b

The equation given is \( y = \frac{k}{x} \). This is the standard form of inverse variation, where the product of \( y \) and \( x \) is a constant \( k \). Hence, it defines inverse variation.

Step 4: Analyze Equation c

The equation is \( y = \frac{x}{k} \), which can be rewritten as \( y = \frac{1}{k}x \). This shows direct variation with \( \frac{1}{k} \) as the constant of proportionality, not inverse variation.

Step 5: Analyze Equation d

The equation is \( d = \frac{k}{g} \). This is similar to inverse variation between \( d \) and \( g \), where the product of \( d \) and \( g \) is constant \( k \). So, it defines inverse variation.

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable changes directly with another. When you see an equation like \( y = kx \), it is a classic example of direct variation. This means: - If one variable increases, the other does too, and vice versa. - The relationship is linear, and any graph would be a straight line passing through the origin. - In this relationship, \( k \) is the constant of proportionality. For example, if you triple \( x \), \( y \) also triples, maintaining the constant ratio defined by \( k \). So, direct variation is all about maintaining a steady proportion between the two variables.

Constant of Proportionality

The constant of proportionality, denoted by \( k \), plays a crucial role in both direct and inverse variations. It represents the constant factor or ratio that links two variables together. - In direct variation, \( k \) appears as the multiplier of \( x \) in the equation \( y = kx \). - In inverse variation, \( k \) is the constant product \( xy = k \). This constant is what keeps the proportional relationship intact: - In direct variation, this could be seen as how much \( y \) changes with respect to a change in \( x \). - In inverse variation, it indicates how \( y \) decreases as \( x \) increases to keep the product \( xy \) consistent.

Variation Equations

Variation equations are mathematical expressions that depict two types of relationships: direct and inverse. These equations help in understanding how one quantity affects another: - **Direct Variation:** The equation form is \( y = kx \), showing a direct relationship where the ratio of \( y \) to \( x \) is constant. - **Inverse Variation:** Represented by \( y = \frac{k}{x} \), where the product of \( y \) and \( x \) remains constant.To determine the type of variation: - Look at how you can rewrite the equation. A form like \( y = kx \) signifies direct variation, while \( y = \frac{k}{x} \) indicates inverse variation. - Variation equations provide an essential foundation for understanding real-world phenomena, such as speed, population growth, or even financial investments, by illustrating how two variables interact with each other.