Problem 2 - Algebra and Trigonometry

Question

If the quantities \(x\) and \(y\) are related by the equation \(y=\frac{3}{x},\) then we say that \(y\) is _____ _____ to \(x\) and the constant of _____ is 3.

Step-by-Step Solution

Answer

y is inversely proportional to x; constant of proportionality is 3.

Step 1: Identify the Type of Relationship

When one quantity depends inversely on another, it is said to be in inverse proportion or inverse variation to that variable. This means that if y depends inversely on x, as x increases, y decreases.

Step 2: Analyze the Given Equation

The equation given is \( y = \frac{3}{x} \). This equation represents an inverse variation since y decreases as x increases, and vice versa.

Step 3: Replace the Blanks

Now, fill in the blanks with the appropriate terms based on the relationship identified in Step 1. The sentence becomes: \( y \) is **inversely proportional** to \( x \) and the constant of **proportionality** is 3.

Key Concepts

Proportionality Constant

The concept of a proportionality constant is crucial in understanding inverse variations. When two quantities are said to be in a mathematical relationship such as inverse proportion, a constant value, known as the proportionality constant, is a vital part of this relationship.
In the equation given, \( y = \frac{3}{x} \), the number 3 is the proportionality constant. This particular constant signifies the degree of variation between two inversely proportional quantities. Whenever you see an expression like \( y = \frac{k}{x} \), you can identify \( k \) as the proportionality constant.

• It signifies that for every unit increase in \( x \), \( y \) will proportionally decrease based on this constant.
• The constant defines the nature and magnitude of the relationship between \( x \) and \( y \).

Thus, understanding the proportionality constant helps in predicting how one variable will react when the other is altered. It plays a pivotal role in distinguishing the specific characteristics of the inverse proportion.

Inverse Proportion

Inverse proportion, also known as inverse variation, defines a relationship where one quantity increases as the other decreases. This scenario is perfectly captured by the equation \( y = \frac{3}{x} \).

• The term "inversely proportional" indicates that the two variables, \( x \) and \( y \), move in opposite directions.
• Here, as \( x \) gets larger, \( y \) becomes smaller.

Inverse proportion is a mathematical relationship where the product of the two variables remains constant. In our example, \( x \times y = 3 \), preserving a constant product.
This concept is widespread in real-life applications, such as the relationship between speed and time for a fixed distance. Understanding inverse proportion allows us to model dynamic systems and predict behavior when variables are adjusted.

Mathematical Relationship

Mathematical relationships describe how two quantities interact with each other. In the given exercise, we see that \( y = \frac{3}{x} \) outlines a specific interaction where \( y \) is dependent on \( x \) in an inverse manner. Such relationships are fundamental in mathematics as they help quantify how changes in one quantity affect another.

• Mathematical relationships can be direct or inverse. This particular example illustrates an inverse relationship.
• Factors like the proportionality constant help define the exact nature of these relationships.

By understanding these relationships, we can better explain and predict the behaviors of different systems. Mathematical relationships are the basis for mathematical modeling used in fields like physics, engineering, economics, and many more. In this case, the equation highlights a clear inverse relationship between \( x \) and \( y \), showcasing how mathematical models can provide insights into the world around us.