Problem 66 - Algebra A Combined Function

Question

Solve. See the Concept Check in this section. Choose the type of variation that each equation represents. a. Direct variation b. Inverse variation c. Joint variation \(x y=\frac{2}{11}\)

Step-by-Step Solution

Answer

The equation represents an inverse variation.

Step 1: Identify the General Form of Variation Equations

Direct variation is typically expressed as \( y = kx \), where \( k \) is a constant. Inverse variation is expressed as \( xy = k \), where \( k \) is a constant. Joint variation involves multiple variables and combinations thereof, for instance, \( z = kxy \). Our goal is to match the given equation to these forms.

Step 2: Analyze the Given Equation

The given equation is \( x y = \frac{2}{11} \). Compare this with the general form of inverse variation, which is \( x y = k \). Here, the product of \( x \) and \( y \) equals a constant, \( \frac{2}{11} \), which directly matches the inverse variation form.

Step 3: Determine the Type of Variation

Since the equation \( x y = \frac{2}{11} \) matches the form of an inverse variation \( xy = k \), the relationship between \( x \) and \( y \) is an inverse variation. In inverse variation, when one variable increases, the other decreases so that their product remains constant (\( \frac{2}{11} \) in this case).

Key Concepts

Inverse Variation

In inverse variation, as one variable increases, the other decreases. This behavior happens in such a way that the product of both variables remains a constant. The fundamental equation for inverse variation is given by \( xy = k \), where \( k \) is a consistent value.

To understand this with an example, if \( x \) increases, \( y \) must decrease so that \( xy \) always equals the constant \( k \). For instance, if \( xy = 4 \), when \( x = 1 \), then \( y = 4 \). Changes in \( x \) must result in corresponding changes in \( y \) to keep the product constant at 4.

• If \( x = 2 \), then \( y = 2 \).
• If \( x = 4 \), then \( y = 1 \).

This relationship showcases how different values of \( x \) and \( y \) still satisfy the condition \( xy = k \). Therefore, in an inverse variation, changes in one variable are balanced by opposite changes in the other while maintaining a consistent product.

Direct Variation

Direct variation describes a linear relationship between two variables. In this context, as one variable increases or decreases, the other variable changes proportionally. The mathematical representation of direct variation is \( y = kx \), where \( k \) is a constant of proportionality.

When dealing with direct variation:

• If \( x \) doubles, \( y \) also doubles.
• If \( x \) is halved, \( y \) is also halved.
• The ratio \( \frac{y}{x} = k \) remains constant.

For example, consider the equation \( y = 3x \):

• If \( x = 2 \), then \( y = 6 \).
• If \( x = 4 \), then \( y = 12 \).

These changes in \( x \) and \( y \) demonstrate a perfectly proportional relationship consistent with the concept of direct variation.

Joint Variation

Joint variation occurs when a variable depends on two or more other variables, typically through multiplication. The typical form of joint variation is expressed as \( z = kxy \), where \( k \) is a constant, and \( z \) varies jointly with both \( x \) and \( y \).

In joint variation:

• All involved variables affect the outcome of \( z \).
• If \( x \) and \( y \) both increase, \( z \) increases as well.
• Keeping one variable fixed and varying the other adjusts \( z \) proportionally to the changing variable.

An example of this would be the equation \( z = 2xy \):

• When \( x = 1 \) and \( y = 1 \), then \( z = 2 \).
• When \( x = 2 \) and \( y = 3 \), then \( z = 12 \).

This shows how changes in \( x \) and \( y \) jointly influence \( z \), with their combined effect being proportional.