Problem 11 - Algebra 2

Question

Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations. $$ \begin{array}{|c|c|c|c|c|}\hline x & {7} & {3} & {1} & {\frac{1}{5}} \\\ \hline y & {\frac{1}{7}} & {\frac{1}{3}} & {1} & {5} \\ \hline\end{array} $$

Step-by-Step Solution

Answer

The relationship is an inverse variation, and the equation that models this variation is \(xy=1\).

Step 1: Identify the Type of Variation

Observing the table, we notice that as the values of \(x\) are decreasing, the values of \(y\) are increasing. Thus, the relationship between \(x\) and \(y\) is not a direct variation but an inverse variation.

Step 2: Formulating Equation for Inverse Variation

An inverse variation relationship can be represented by the equation \(xy=k\), where \(k\) is the constant of variation. To find the constant of variation, multiply a pair of \(x\) and \(y\) values. Choosing \(x=7\) and \(y=1/7\), the product is \(7*(1/7)=1\). So, the constant of variation, \(k\), is 1.

Step 3: Write the Equation for the Inverse Variation

From the previous step, we know the constant of variation (\(k\)) is 1. Substituting \(k\) into the equation of inverse variation \(xy=k\), we get: \(xy=1\).

Key Concepts

Direct Variation

In mathematics, a **direct variation** refers to a relationship between two variables, where one variable is a constant multiple of the other. This means that as one variable increases, the other also increases proportionally. The equation that represents direct variation is expressed as \( y = kx \), where \( k \) is the constant of variation. In a direct variation scenario, the graph of the equation would be a straight line passing through the origin (0,0).

• Example: If \( y = 3x \), then \( y \) directly varies with \( x \), as multiplying \( x \) by the constant 3 yields \( y \).
• In the exercise provided, the relationship is not of direct variation because the values of \( y \) do not change in proportion to \( x \). Instead, they follow an inverse pattern.

Understanding direct variation helps in identifying when changes in one variable directly influence similar changes in another variable.

Constant of Variation

The **constant of variation** is a crucial concept in understanding the relationship between variables in both direct and inverse variation. It's a factor that remains consistent when variables change. In direct variation, it's the multiplier used in the equation \( y = kx \), and in inverse variation, it appears in the equation \( xy = k \).

Calculating the constant of variation is key. To find \( k \), you can use any given values from the equation. For example, in inverse variation, if you know \( x = 7 \) and \( y = \frac{1}{7} \), the constant is computed as \( k = 7 \times \frac{1}{7} = 1 \).

• It acts as the anchor that links the variables together.
• In any variation equation, knowing \( k \) allows you to predict one variable based on another.

For the problem discussed, the constant of variation \( k \) was identified as 1, highlighting the inverse relationship.

Mathematical Modeling

**Mathematical modeling** is a method of representing real-world situations using mathematical expressions and equations to analyze and make predictions. In this context, it involves identifying the type of variation and formulating the correct equation.

For this exercise, mathematical modeling helped us determine that the variable relationship was one of inverse variation. This conclusion arose from observing the behavior of \( x \) and \( y \). Whenever \( x \) decreased, \( y \) increased, suggesting an inverse relationship.

• Mathematical models provide frameworks to predict outcomes.
• They allow translation of data relationships into mathematical terms.
• Such models are foundational in fields like physics, economics, and engineering.

Thus, correctly identifying the model as \( xy = k \) is essential for analysis and solving challenges similar to the exercise.

Equation Formulation

Formulating equations is a fundamental skill in mathematics that helps in defining the relationship between different variables. For this exercise, formulating the equation correctly allowed us to express the inverse relationship between \( x \) and \( y \).

Starting with identifying the type of variation, we then set out to find the constant of variation \( k \). Once \( k \) was determined, we used it to write the inverse variation equation: \( xy = 1 \).

• Correct equation formulation is critical. It ensures the relationship is accurately depicted.
• The process involves using known data points and logical inference.
• Equations become the bridge between theoretical understanding and practical application.

Being able to create this equation means we can now predict one variable if the other is known, demonstrating the power of mathematics in describing relationships.