Problem 6 - College Algebra

Question

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(w\) varies inversely as \(l\)

Step-by-Step Solution

Answer

The equation that expresses the relationship where \(w\) varies inversely as \(l\) is \(w = \frac{k}{l}\).

Step 1: Understanding inverse variation

An inverse variation between two variables is represented mathematically as \(y = \frac{k}{x}\), where \(k\) is the constant of variation. In this exercise, \(w\) varies inversely as \(l\). Therefore, \(w\) and \(l\) are in the positions of \(y\) and \(x\) respectively.

Step 2: Writing the equation

Following the inverse variation formula and replacing \(w\) and \(l\) for \(y\) and \(x\) respectively and \(k\) is the constant of variation, we obtain the equation: \(w = \frac{k}{l}\).

Step 3: Final Expression

The final equation that expresses the given relationship where 'w' varies inversely as 'l' is \(w = \frac{k}{l}\). This equation tells us that as \(l\) increases, \(w\) decreases in proportion, and vice versa, with \(k\) being the constant of variation.

Key Concepts

Constant of Variation

In the realm of inverse variation, the constant of variation, often symbolized as \(k\), plays an essential role. This constant captures the unique relationship between two variables, indicating how one changes in response to the other. Notably, \(k\) remains constant throughout the relation, despite fluctuations in the variables themselves.
If you think about the equation \(w = \frac{k}{l}\), where \(w\) and \(l\) are inversely related, \(k\) represents the product of \(w\) and \(l\).

• When \(w\) increases, \(l\) must decrease to keep the product equal to \(k\).
• Conversely, if \(l\) grows, \(w\) will shrink accordingly.

In essence, the constant of variation is that fixed product which helps us predict and understand the behavior of \(w\) relative to \(l\).

Variables

Variables are foundational in expressing mathematical relationships, acting as placeholders that can take on different values. In our inverse variation context, we have two specific variables: \(w\) and \(l\).

• \(w\): This is the dependent variable, meaning its value depends on the change in \(l\).
• \(l\): This is the independent variable that dictates the values of \(w\) based on its fluctuations.

It is crucial to identify these roles when writing the corresponding equation, as mistaking their roles can lead to incorrect interpretations and predictions of the relationship.

Relationship Equation

The heart of inverse variation lies in its relationship equation, \(w = \frac{k}{l}\), which succinctly captures how two variables are interconnected. This equation tells us that \(w\) is inversely proportional to \(l\).

• If \(l\) were to double, \(w\) would be halved, illustrating the inverse relationship.
• Similarly, a reduction in \(l\) causes a corresponding increase in \(w\).

By analyzing this equation, students can predict the behavior of one variable based on the given or assumed values of the other and the constant \(k\), fostering a deeper understanding of inverse relationships.

Algebraic Expressions

Algebraic expressions are the language of mathematics, used to represent how quantities relate to one another. Our current expression, \(w = \frac{k}{l}\), exemplifies how inverse variation can be captured using algebra.

• "Inverse" implies that as one variable increases, the other decreases, which is clearly seen in the equation's structure.
• The fraction format highlights the division aspect, central to understanding inverses.

Through this expression, we not only see the connection between \(w\) and \(l\), but it also gives us a way to compute \(w\) for any given \(l\) and constant \(k\), bringing algebra's power to light in interpreting variable relationships.