Problem 18 - Intermediate Algebra

Question

Find the constant of variation for each of the stated conditions. \(y\) varies inversely as \(x\), and \(y=-6\) when \(x=\frac{4}{3}\).

Step-by-Step Solution

Answer

The constant of variation is -8.

Step 1: Understand Inverse Variation

When a variable y varies inversely as another variable x, it means that \( y \times x = k \), where \( k \) is the constant of variation. This relationship tells us that the product of the two variables remains constant.

Step 2: Substitute the Known Values

We know from the problem that \( y = -6 \) when \( x = \frac{4}{3} \). Substitute these values into the inverse variation formula: \( -6 \times \frac{4}{3} = k \).

Step 3: Calculate the Constant of Variation

Calculate \( k \) by multiplying the known values: \(-6 \times \frac{4}{3} = -8\). This means that the constant of variation \( k \) is \(-8\).

Key Concepts

Constant of Variation

In an inverse variation, the constant of variation is a key element. It is denoted as \( k \) and represents the product of two inversely related variables. When solving problems in inverse variation, the formula \( y \times x = k \) is used. Here, \( y \) and \( x \) are the variables, while \( k \) stays constant. No matter how \( y \) and \( x \) change, their product \( k \) remains the same.

For example, if \( y = -6 \) and \( x = \frac{4}{3} \), the constant of variation can be found by calculating \( -6 \times \frac{4}{3} \). This results in \( k = -8 \). This value signifies the constant linkage between the variables in this particular inverse variation scenario.

Inverse Relationship

An inverse relationship is when two variables move in opposite directions, such that as one increases, the other decreases. This type of relationship is common in various mathematical contexts.

Here is how it works:

• When \( y \) varies inversely as \( x \), larger values of \( x \) will result in smaller values of \( y \), and vice versa.
• This relationship is captured in the equation \( y \times x = k \), clarifying that as one variable increases, the other must decrease to keep the product \( k \) consistent.

Understanding inverse relationships is crucial as it unveils how variables respond to changes in one another. This principle is not only vital for solving equations but also for comprehending real-world phenomena.

Mathematics Education

Understanding concepts like inverse variation is vital in mathematics education, as they form foundational knowledge required for advanced topics. Using concrete examples helps solidify these abstract concepts.

Here are some common educational approaches used to teach inverse variation and similar topics:

• Problem-Solving: Tackling specific problems, like finding the constant of variation, enhances learning by applying theory to practical scenarios.
• Visual Aids: Graphs and charts can help visualize how two variables relate inversely, giving students a clearer picture of the concept.
• Interactive Learning: Engaging in group exercises and discussions encourages students to explore inverse relationships further and retain information effectively.

Such methods help in internalizing the principle that as one variable grows, the other shrinks, a cornerstone of understanding inverse variation.