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📈college algebra review

Varies Directly With

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

The term 'varies directly with' describes a relationship between two variables where one variable changes in direct proportion to the other. This means that as one variable increases, the other variable increases at the same rate, and vice versa. This concept is fundamental to understanding the topic of Modeling Using Variation covered in chapter 5.8.

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5 Must Know Facts For Your Next Test

  1. When two variables vary directly, their ratio remains constant, and they can be represented by the equation $y = kx$, where $k$ is the constant of proportionality.
  2. Direct variation is often used to model real-world situations, such as the relationship between the cost of an item and the quantity purchased, or the relationship between the speed of a vehicle and the distance traveled.
  3. The constant of proportionality, $k$, represents the rate of change between the two variables and can be determined by dividing one variable by the other.
  4. Direct variation is a linear relationship, meaning that a graph of the two variables will be a straight line passing through the origin.
  5. Understanding direct variation is crucial for solving problems related to proportional reasoning, scaling, and other applications of linear relationships.

Review Questions

  • Explain how the equation $y = kx$ represents a direct variation relationship between the variables $x$ and $y$.
    • The equation $y = kx$ represents a direct variation relationship because it shows that the variable $y$ changes in direct proportion to the variable $x$. The constant $k$ is the constant of proportionality, which means that as $x$ increases by a certain amount, $y$ increases by the same proportional amount. This indicates that the ratio between $x$ and $y$ remains constant, a defining characteristic of a direct variation relationship.
  • Describe how the graph of a direct variation relationship would appear and explain the significance of the constant of proportionality, $k$, in the graph.
    • The graph of a direct variation relationship would be a straight line passing through the origin. The constant of proportionality, $k$, would determine the slope of the line. Specifically, the value of $k$ would represent the rate of change between the two variables, meaning that for every unit increase in the independent variable, the dependent variable increases by $k$ units. The constant of proportionality, $k$, is a crucial parameter in direct variation relationships as it quantifies the strength of the linear relationship between the two variables.
  • Analyze how the concept of direct variation can be applied to solve real-world problems, and provide an example to illustrate its usefulness.
    • The concept of direct variation can be applied to solve a wide range of real-world problems, particularly those involving proportional reasoning. For example, the relationship between the cost of an item and the quantity purchased often exhibits direct variation. If the cost per item remains constant, then as the quantity purchased increases, the total cost will increase proportionally. Similarly, the relationship between the speed of a vehicle and the distance traveled can be modeled using direct variation, where the distance traveled is directly proportional to the speed and the time elapsed. Understanding direct variation allows us to make accurate predictions, scale measurements, and solve problems involving proportional relationships in various contexts, making it a valuable tool in applied mathematics and science.