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Chapter 8: Problem 69

Translate each statement into an equation. \(x\) varies directly as \(y\)

Short Answer

Expert verified
The equation is \(x = ky\).

Step by step solution

01

Identify the Direct Variation

In direct variation problems, we have a relationship between two variables where one variable is a constant multiple of the other. If we say that \(x\) varies directly as \(y\), it means that as \(y\) changes, \(x\) changes proportionally. We need to find a way to express this relationship mathematically.
02

Write the Direct Variation Equation

When \(x\) varies directly as \(y\), it implies a relationship that can be written as \(x = ky\), where \(k\) is a constant of proportionality. This equation represents the direct variation between \(x\) and \(y\).

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Key Concepts

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

Constant of Proportionality
In direct variation, the constant of proportionality is crucial. It defines the factor by which one variable changes in relation to another. Consider the equation \( x = ky \). Here, \( k \) is the constant of proportionality. It remains unchanged as both \( x \) and \( y \) vary. This constant helps us precisely determine the multiplier effect of \( y \) on \( x \). For example, if \( k = 3 \), then for every unit of \( y \), \( x \) increases by 3 units.
  • The constant allows us to predict how much one variable changes when the other does.
  • It remains the same even if the values of \( x \) and \( y \) change, maintaining proportionality.
Understanding \( k \) helps not only in finding a relationship but also in solving real-world problems involving direct variation.
Variables
Variables are symbols used to represent numbers or values that can change within a given problem. In the context of direct variation, we primarily work with two variables, often represented as \( x \) and \( y \).
  • \( x \) is typically the dependent variable, meaning that its value depends on \( y \).
  • \( y \) is generally the independent variable, the one we have control over or can set freely.
In our equation \( x = ky \), the behavior of \( x \) depends on the values assigned to \( y \) and the constant \( k \). Variables provide flexibility and are essential in expressing mathematical relationships clearly and concisely, adapting the equation for various practical scenarios.
Equation Translation
Translating statements into equations is an important step in solving mathematical problems. It involves converting a verbal phrase into a mathematical expression or equation.
For instance, in the statement, "\( x \) varies directly as \( y \)," we translate it to the equation form \( x = ky \). This translation helps us understand and manipulate mathematical relationships more effectively.
  • Using equations simplifies complex relationships into understandable mathematical terms.
  • The process requires identifying the relationship (direct, indirect, etc.) and expressing it accurately.
Mastering equation translation is key to solving algebra problems, as it lays the foundation for further mathematical computation and reasoning.

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