Problem 38 - Introductory Chemistry Atoms First

Question

Suppose the variable \(x\) is proportional to \(1 / y\). What does this tell you about how the numeric value of \(x\) changes as the numeric value of \(y\) changes?

Step-by-Step Solution

Answer

In conclusion, when \(x\) is proportional to \(\frac{1}{y}\), it means that \(x\) and \(y\) are inversely related. As the numeric value of \(y\) increases, the numeric value of \(x\) decreases, and vice versa. Their relationship can be represented using the equation: \(x = k\left(\frac{1}{y}\right)\), where \(k\) is the constant of proportionality.

Step 1: Understand Proportionality

Proportionality means that one variable is directly related to another by a constant factor. In our case, \(x\) is proportional to \(\frac{1}{y}\). We can represent this relationship using the following equation: \[x = k\left(\frac{1}{y}\right)\] where \(k\) is the constant of proportionality.

Step 2: Examine the Relationship Between \(x\) and \(y\)

Now let's analyze how the numeric value of \(x\) changes as the numeric value of \(y\) changes. Since \(x\) is proportional to \(\frac{1}{y}\), if \(y\) increases, \(\frac{1}{y}\) decreases and if \(y\) decreases, \(\frac{1}{y}\) increases. This relationship can be verified by considering the equation: \[x = k\left(\frac{1}{y}\right)\] When the value of \(y\) increases, the value of \(\frac{1}{y}\) decreases, leading to a decrease in the value of \(x\). Conversely, when the value of \(y\) decreases, the value of \(\frac{1}{y}\) increases, leading to an increase in the value of \(x\).

Step 3: Conclusion

In conclusion, the numeric value of \(x\) is inversely proportional to the numeric value of \(y\). As \(y\) increases, \(x\) decreases, and as \(y\) decreases, \(x\) increases. This relationship can be represented by the equation: \[x = k\left(\frac{1}{y}\right)\] where \(k\) is the constant of proportionality.

Key Concepts

Understanding Constants

In mathematics, a constant is a fixed value that does not change. It is a specific number or symbol representing a fixed idea or quantity. In the context of proportional relationships, we often refer to a constant of proportionality. This constant helps us understand how two variables are related.
When we say a variable is inversely proportional to another, like in our example where \( x \) is proportional to \( \frac{1}{y} \), there is a constant \( k \) involved. The formula \( x = k \left(\frac{1}{y}\right) \) clearly shows how the constant \( k \) plays a role. Here:

• \( k \) remains the same no matter what values \( x \) or \( y \) take.
• \( k \) provides a baseline to explain the degree of change between \( x \) and \( y \).

Without \( k \), it would be challenging to capture the unique relationship between variables. It anchors the equation, ensuring that no matter how \( y \) changes, the relationship between \( x \) and \( y \) remains consistent.

The Role of Variables

Variables are symbols used to represent values that can change or vary. In equations, they help us explore and express relationships between different quantities. For example, in the equation \( x = k \left( \frac{1}{y} \right) \), both \( x \) and \( y \) are variables.
Understanding the behavior of variables is crucial in studying inverse proportionality. Here is how variables \( x \) and \( y \) interact in our example:

• When \( y \) increases, the expression \( \frac{1}{y} \) becomes smaller, which in turn makes \( x \) smaller.
• Conversely, when \( y \) decreases, \( \frac{1}{y} \) becomes larger, resulting in a larger value for \( x \).

This connection illustrates how variables help us build a dynamic picture of mathematical relationships. Variables adjust based on each other's changes, while constants maintain the stability of the relationship.

Forming Mathematical Equations

Mathematical equations are expressions that define relationships between numbers and variables. They are the language of mathematics, used to convey precise information. Equations like \( x = k \left( \frac{1}{y} \right) \) help us explain how entities like \( x \) and \( y \) are connected.
In inverse proportionality, equations serve several purposes:

• They succinctly describe complex relationships. For instance, \( x = k \left( \frac{1}{y} \right) \) is a clear statement of how \( x \) inversely relates to \( y \).
• They allow us to predict values. For example, given \( k \) and one variable, we can solve for the other variable.
• They help in visualizing patterns, which can be useful for graphing and further analysis.

Mathematical equations are crucial tools for understanding and analyzing how different quantities affect each other. They provide a consistent method for solving problems and exploring mathematical relationships. In our case, it aids in grasping the concept of inverse proportionality clearly.