Problem 35 - Algebra for College Students

Question

Suppose that \(y\) varies inversely as \(x\). Does doubling the value of \(x\) also double the value of \(y\) ? Explain your answer.

Step-by-Step Solution

Answer

No, doubling \( x \) halves \( y \) in inverse variation.

Step 1: Understand Inverse Variation

In an inverse variation relationship, the product of two variables is constant. This means that if \( y \) varies inversely as \( x \), then \( y = \frac{k}{x} \), where \( k \) is a constant.

Step 2: Analyze Change in \( x \)

If \( x \) is doubled, then the new value of \( x \) becomes \( 2x \). We need to determine how this affects \( y \).

Step 3: Calculate New Value of \( y \)

Substitute \( 2x \) for \( x \) in the inverse variation equation: \( y = \frac{k}{2x} \). This implies the new \( y \) becomes \( \frac{1}{2} \times \frac{k}{x} \), or half the original value of \( y \).

Step 4: Conclusion on \( y \) Change

Doubling \( x \) results in \( y \) being halved because \( y = \frac{k}{x} \) implies \( y \) will decrease as \( x \) increases to maintain their product constant.

Key Concepts

Inverse Proportion

Inverse proportion is a relationship between two variables where their product remains constant. Think of it as a balancing act: if one variable increases, the other must decrease to keep the balance. This is expressed mathematically as \( y = \frac{k}{x} \), where \( k \) is a constant and \( y \) varies inversely with \( x \).
If we consider a seesaw, imagine one side going up when the other goes down. This visual helps understand how, in inverse proportion, increasing one variable forces the other to diminish. In our original exercise, increasing \( x \) led to a smaller value of \( y \), illustrating this see-saw effect.

• Inverse proportion relationships are commonly found in real life, like speed and time for a journey.
• Understanding this concept is key in solving problems where maintaining a constant product is required.

Mathematical Constant

The notion of a mathematical constant plays a vital role in coping with inverse proportions. In the equation \( y = \frac{k}{x} \), \( k \) signifies a constant—a number that remains the same throughout the variation process. This constant represents the unchanging product of the two inversely proportional variables.
When dealing with exercises involving inverse variation, identifying and maintaining this constant is crucial. Because no matter how \( x \) or \( y \) change, \( k \) remains unfazed, allowing for the predictability and reliability in calculations.

• Mathematical constants simplify complex relationships by providing a stable reference.
• They are foundational in diverse mathematical scenarios beyond just inverse variations.

Variable Dependence

In inverse proportion, variable dependence illustrates how changes in one variable affect the other. Specifically, the inverse relationship is characterized by one variable's rise leading to a fall in the other, to keep the product constant.
This dependence is crucial—it suggests a predictable and calculable change. In the given exercise, doubling \( x \) reduces \( y \) by half, neatly showcasing this dependence. Understanding this interaction helps predict outcomes when altering variables in inverse relations.

• Variable dependence guides the manipulation of equations by highlighting how to maintain balance.
• It underlines the principle that changing one variable necessitates compensation in the other to uphold the constant.