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Chapter 7: Problem 9

Determine whether each equation represents direct, inverse, joint, or combined variation. $$ y=\frac{3}{x} $$

Short Answer

Expert verified
Inverse variation.

Step by step solution

01

Identify the given equation

The given equation is \( y = \frac{3}{x} \).
02

Understand variation types

Direct variation is in the form \( y = kx \), inverse variation is in the form \( y = \frac{k}{x} \), joint variation combines direct variations of several variables, and combined variation combines direct and inverse variations.
03

Match the given equation to variation types

Compare \( y = \frac{3}{x} \) with the general form of inverse variation, which is \( y = \frac{k}{x} \). The given equation fits this form, with \( k = 3 \).
04

Determine the type of variation

Since the given equation is in the form \( y = \frac{k}{x} \), it represents an inverse variation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Direct Variation
Direct variation is a straightforward relationship between two variables in which one variable is a constant multiple of the other. This relationship is expressed mathematically as \( y = kx \), where \( y \) and \( x \) are variables, and \( k \) is a non-zero constant.
In direct variation:
  • When \( x \) increases, \( y \) also increases.
  • When \( x \) decreases, \( y \) also decreases.
  • The ratio \( \frac{y}{x} \) is always equal to the constant \( k \).
Direct variation is often found in real-world scenarios such as speed and distance. For instance, if you drive at a constant speed, the distance traveled varies directly with the time spent driving.
Joint Variation
Joint variation occurs when a variable is directly proportional to the product of two or more other variables. The general form of joint variation is \( z = kxy \), where \( z \), \( x \), and \( y \) are variables, and \( k \) is a constant.
In joint variation:
  • As either \( x \) or \( y \) increases, \( z \) increases if the other variable remains constant.
  • As both \( x \) and \( y \) increase together, \( z \) increases more significantly.
An example of joint variation can be seen in physics, where the force (\f \( F \)) required to compress a spring is jointly proportional to the displacement (\f \( x \)) and a constant (\f \( k \)). The formula is \[ F = kx \].
Combined Variation
Combined variation involves a relationship that includes both direct and inverse variations within the same equation. Typically, it is expressed in forms like \( y = k \frac{x}{z} \) or \( yz = kx \), where \( y \), \( x \), and \( z \) are variables, and \( k \) is a constant.
In combined variation:
  • One part of the variable relationship might increase while another might decrease.
  • Combined variations are more complex because they encompass multiple types of relationships within a single equation.
For example, the amount of work done might vary directly with the number of workers and inversely with the time taken. This can be represented as \( W = k \frac{N}{T} \), where \( W \) is the work, \( N \) is the number of workers, and \( T \) is the time.

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