Question

CRITICAL THINKING Suppose \(x\) varies inversely with \(y\) and \(y\) varies inversely with \(z\). How does \(x\) vary with \(z\) ? Justify your answer.

Short answer

In conclusion, if \(x\) varies inversely with \(y\) and \(y\) varies inversely with \(z\), then \(x\) varies directly with \(z\).

Step-by-step solution

Represent the inverse variations

Since \(x\) varies inversely with \(y\), this could be written as \(xy = k_1\), where \(k_1\) is the constant of variation between \(x\) and \(y\). Similarly, since \(y\) varies inversely with \(z\), this can be expressed as \(yz = k_2\), where \(k_2\) represents the constant of variation between \(y\) and \(z\).

Solve for y in inverse relations

Re-arrange the first equation \(xy = k_1\) to solve for \(y\), which yields, \(y = k_1/x\). Similarly, from the equation \(yz = k_2\), solving for \(y\), we get \(y = k_2/z\). The resulting equations help to later substitute and solve for a direct relation between \(x\) and \(z\).

Equate the equations and simplify

Since from both equations, \(y = k_1/x\) and \(y = k_2/z\), \(y\) is a common variable, equate the right hand sides, we have \(k_1/x = k_2/z\). This could be rearranged to the form, \(xz = k_1/k_2\). Thus confirming that \(x\) varies directly with \(z\).

Key concepts

Direct Variation

Direct variation is when two variables change in such a manner that a change in one variable causes a proportional change in the other.
This relationship can be represented by the equation \( y = kx \), where \( k \) is the constant of variation. Essentially, when one variable increases, the other also increases proportionally, and when one decreases, the other does too.
When we talk about direct variation in mathematics, it helps to visualize this as a straight line that passes through the origin if plotted on a graph.

• If \( x \) doubles, \( y \) doubles.
• If \( x \) is halved, \( y \) is also halved.

This proportional relationship is crucial for understanding how variables relate to each other directly.
In the context of the exercise, understanding direct variation allowed us to conclude that \( x \) and \( z \) have a linear relationship once the inversions are calculated.

Constants of Variation

The constant of variation is a key part of both direct and inverse variation.
In an equation that expresses a variation, the constant of variation represents the fixed ratio or product of the variables involved.
In direct variation (like \( y = kx \)), \( k \) represents this constant factor by which one variable multiplies to get the other.
For inverse variation, this constant ensures whatever happens with one variable, an opposite happens with the other. For instance, if \( x \) and \( y \) are inversely related through the equation \( xy = k \), the value of \( k \) remains consistent regardless of the specific values of \( x \) and \( y \) as long as they're inversely related.

• In the given exercise, the constant \( k_1 \) relates \( x \) and \( y \), while \( k_2 \) relates \( y \) and \( z \).

Beyond just numbers, these constants allow us to build meaningful connections between variables and deepen our understanding.

Mathematical Reasoning

Mathematical reasoning is the process of finding a logical trail from assumptions to consequent conclusions.
This kind of reasoning helps us understand, solve, and even verify mathematical equations and their relationships.
In the original exercise, we utilized reasoning skills to determine how \( x \) and \( z \) relate. By carefully examining the inverse relationships, we equated separate expressions to uncover a direct relationship between \( x \) and \( z \).

• It's about comparing the left and right of the equation.
• It requires connecting dots logically.

The key lays in understanding each step's logic in your mathematical journey.
With practice, mathematical reasoning becomes intuitive, guiding you through more complex problems with ease and clarity.