Problem 36 - Precalculus with Limits

Question

DIRECT VARIATION In Exercises 35-38, assume that is \(y\) directly proportional to \(x\). Use the given \(x\)-value and \(y\)-value to find a linear model that relates \(y\) and \(x\). \(x = 2\), \(y = 14\)

Step-by-Step Solution

Answer

The linear model that relates \(y\) and \(x\) is \(y = 7x\).

Step 1: Identify the Given Values

We are given that \(x = 2\) and \(y = 14\). We will use these values to find the value of the constant of proportionality \(k\).

Step 2: Apply the values to the equation

We substitute \(x = 2\) and \(y = 14\) into the equation \(y = kx\). This gives us the equation \(14 = k * 2\).

Step 3: Solve for the constant \(k\)

To solve for \(k\), we divide both sides of the equation by 2. This gives us \(k = 14 / 2\) or \(k = 7\).

Step 4: Find the Linear Model

The linear model that relates \(y\) and \(x\) is given by \(y = 7x\).

Key Concepts

Linear Model

In mathematics, especially in the context of direct variation, a linear model is an equation that describes the relationship between two variables using a straight line. The formula is often written as \( y = kx \) where \( y \) is the dependent variable, \( x \) is the independent variable, and \( k \) is the constant of proportionality. The key idea behind a linear model is that it shows a consistent, linear relationship between overall changes in \( x \) and \( y \). Some features of a linear model include:

• Every change in \( x \) results in a proportionate change in \( y \).
• The graph of the equation is a straight line passing through the origin (0,0).
• This model is deterministic, meaning it predicts outcomes exactly and without randomness within the defined constraints.

Linear models are especially useful for making predictions and understanding how one variable affects another in a linear way.

Constant of Proportionality

The constant of proportionality, \( k \), is a crucial component of direct variation relationships. This constant determines the rate at which one variable changes with respect to another in the equation \( y = kx \).The process of finding \( k \) involves:

• Identifying given values of \( x \) and \( y \).
• Substituting these values into the equation \( y = kx \).
• Solving for \( k \) by isolating it on one side of the equation. For example, if \( y = 14 \) and \( x = 2 \), then plugging into the formula gives \( 14 = 2k \), translating to \( k = 7 \).

Understanding \( k \) is fundamental because it influences the slope of the line in the linear model, indicating how steep the graph will be and how strongly \( x \) and \( y \) are connected.

Proportional Relationship

A proportional relationship forms when two variables maintain a constant ratio. In contexts such as direct variation, this implies that as one variable increases or decreases, the other does so at a consistent rate, dictated by the constant of proportionality.Characteristics of proportional relationships include:

• The formula employed is generally \( y = kx \), showcasing a direct link where \( k \) is the constant coefficient.
• Graphically represented as a straight line through the origin, reinforcing the idea of direct proportionality and uniform change.
• These relationships maintain consistency, meaning doubling one variable results in doubling the other if \( k \) remains unchanged.

Proportional relationships are seen everywhere in real-world applications, from scaling recipes to understanding speed and time dynamics in physics. Recognizing these relationships can simplify complex problems by focusing on proportional reasoning.