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Chapter 1: Problem 20

Direct Variation In Exercises \(19-26,\) assume that \(y\) is directly proportional to \(x .\) Use the given \(x\) -value and \(y\) -value to find a linear model that relates \(y\) and \(x .\) $$x=5, y=12$$

Short Answer

Expert verified
The linear model that relates \(y\) and \(x\) is \(y = 2.4x\).

Step by step solution

01

Identify relation and given values

Given the relation \(y\) is directly proportional to \(x\), we use the proportionality equation \(y = kx\). We are also given the values \(x = 5\) and \(y = 12\).
02

Substitute and solve for k

Rearrange the proportionality equation to solve for \(k\) as follows: \(k = y / x\). Substitute the given values, \(y = 12\) and \(x = 5\), into the equation. We get \(k = 12 / 5 = 2.4\). So, the constant of variation, \(k\), is 2.4.
03

Write down the linear model

Now that we have the constant of variation, we return to the original proportionality equation, \(y = kx\), and substitute \(k\) with its value. We thus obtain the final linear model: \(y = 2.4x\).

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Key Concepts

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

Understanding the Linear Model in Direct Variation
Direct variation is a straightforward mathematical concept where one variable changes in direct proportion to another. This means that as one variable increases or decreases, the other does so as well, by a constant factor. This relationship forms what we call a *linear model*. In our exercise, the linear model is given by the equation \( y = kx \). This simple equation tells us that the value of \( y \) is always \( k \) times whatever \( x \) is.

Key Points about Linear Models:
  • The graph of a direct variation relationship is a straight line passing through the origin \((0, 0)\).
  • The linear model is helpful for predicting one variable when you know the other.
  • The slope of the line, \( k \), tells you how steep or flat the line is.
Understanding these properties of linear models can make analyzing and predicting data much easier. In the context of the exercise, we used this model to find the relationship between \( x \) and \( y \).
Identifying the Proportionality Constant
A crucial part of working with direct variation is finding the *proportionality constant*, often denoted by \( k \). The proportionality constant is the fixed number that describes the rate at which \( y \) changes concerning \( x \). In our exercise, we identified \( k \) by using the equation \( k = \frac{y}{x} \), meaning you divide \( y \) by \( x \).

Steps to Find the Proportionality Constant:
  • Use the known values of \( x \) and \( y \) to plug into the equation \( k = \frac{y}{x} \).
  • Calculate \( k \) to find the rate of change. For example, with \( x = 5 \) and \( y = 12 \), we found \( k = 2.4 \).
  • Substitute \( k \) back into the linear model equation to fully understand how \( x \) and \( y \) relate.
Determining \( k \) is essential because it is the core of the direct variation relationship. Without it, we cannot construct our linear equation, or accurately predict how changing \( x \) will change \( y \).
Solving Equations for Direct Variation
Solving equations in the context of direct variation is about finding unknown values using the relationship \( y = kx \). Once we have our proportionality constant, \( k \), fitting \( x \) or \( y \) into this relationship becomes straightforward. The main aim is to solve for whatever variable is missing.

Steps in Solving Direct Variation Equations:
  • Identify all known values and which variable you need to solve for. From our example, given \( x = 5 \) and \( y = 12 \), \( k \) was first determined.
  • Insert the known values into the equation \( y = kx \) and rearrange to solve for the unknown. If \( y \) is missing, solve \( y = kx \), and vice-versa.
  • Check the solution by ensuring it satisfies the original condition of direct proportionality.
By approaching these equations step by step, you ensure clarity and maintain the direct proportionality, making these problems much simpler to solve. Remember, the direct variation context keeps solutions clean and linear, which helps immensely in avoiding complexity.

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Most popular questions from this chapter

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