Problem 99 - College Algebra with Modeling and Visualization

Question

Assume that the constant of proportionality is positive. Let \(y\) be inversely proportional to \(x\). If \(x\) doubles, what happens to \(y ?\)

Step-by-Step Solution

Answer

If \(x\) doubles, \(y\) becomes half its initial value.

Step 1: Understanding Inverse Proportionality

When we say that a variable \(y\) is inversely proportional to another variable \(x\), it means that the product of \(x\) and \(y\) is a constant. This can be expressed as \(xy = k\) where \(k\) is the constant of proportionality.

Step 2: Initial Equation

Given the inverse proportionality, write the initial relationship as \(xy = k\). If initially \(x = x_0\) and \(y = y_0\), then \(x_0 y_0 = k\).

Step 3: Doubling \(x\)

Now, we consider what happens when \(x\) doubles. Let \(x\) change to \(2x_0\). We need to find the new value of \(y\) that we'll call \(y_1\) where \(2x_0 \cdot y_1 = k\).

Step 4: Solving for New \(y\)

Since the product \(xy = k\) stays constant, substitute \(k\) from the initial equation: \(x_0 y_0 = 2x_0 y_1\). Solving for \(y_1\), divide both sides by \(2x_0\), giving \(y_1 = \frac{y_0}{2}\).

Step 5: Conclusion

When \(x\) doubles, \(y\) becomes half of its initial value. Thus, \(y\) is inversely affected by the change in \(x\).

Key Concepts

Constant of Proportionality

The constant of proportionality is a vital concept when dealing with relationships between two variables. Imagine it like a glue that holds the relationship together. Specifically, in the context of inverse proportionality, when one variable increases, the other one decreases, and vice versa.
This is because the product of the two variables remains constant. This is expressed mathematically as \( xy = k \), where \( k \) is the constant of proportionality.
When we talk about inverse relationships, if you know the value of the constant, you can predict how changes in one variable affect the other. For instance, if you have a smaller value for \( k \), both variables must have relatively lower values for their product to equal \( k \).
On the other hand, a larger \( k \) indicates possibly larger values for the variables involved, still keeping their product as \( k \). Here, constancy is key; no matter what the values of \( x \) and \( y \) are, their multiplication will always yield this constant.

Algebraic Relationships

Algebraic relationships provide a structured way to manage different variables and their connections. In case of inverse proportionality, the relationship between \( x \) and \( y \) is expressed via the equation \( xy = k \). This equation is at the heart of connecting these variables together.
Algebra helps dissect these connections and predict changes within the system. For example, if you know \( k \) and one variable's value, you can easily solve for the other variable. Rearrange the equation \( y = \frac{k}{x} \) to see the influence each variable has on the other.
This understanding of algebraic relationships enables you to analyze problems involving inverse proportionality easily. You parse through changes systematically, ensuring that the constancy of \( k \) is upheld at all times.
When you become familiar with these relationships, they remove the mystery around variable interactions. You learn how to manipulate equations to find desired outcomes or predict effects of variable adjustments.

Variable Manipulation

Variable manipulation is a powerful tool when dealing with equations, especially ones involving proportionality. By "manipulating," we mean altering the variables in structured ways to gain more insight or to reach a solution.
When \( y \) is inversely proportional to \( x \) (meaning \( xy = k \)), every change in \( x \) directly impacts \( y \). For instance, doubling \( x \) sets off a change, requiring you to recalibrate \( y \) for the equation to hold true. With \( 2x_0 \), to keep \( xy = k \), \( y \) must adapt so that \( y = \frac{y_0}{2} \).
Understanding and accurately manipulating variables in such equations ensures the system remains intact despite changes. It requires careful calculation and balance; otherwise, you risk misrepresenting the relationship. Regular practice in manipulating and solving these variables not just solves problems but enhances overall algebraic fluency.
Ultimately, variable manipulation means you become adept at maintaining the integrity of relationships within equations, thereby solving complex real-world problems with relative ease.