Problem 60 - Algebra 2

Question

Which equation does NOT represent inverse variation between \(x\) and \(z ?\) $$ \begin{array}{ll}{\text { A. } x=\frac{y}{z}} & {\text { B. } x=\frac{-15 z}{y}} \\ {\text { C. } z=\frac{-15 y}{x}} & {\text { D. } x z=5 y}\end{array} $$

Step-by-Step Solution

Answer

As this exercise is structured, there are actually three correct answers: A, B, and C. All these options do not represent an inverse variation between \(x\) and \(z\).

Step 1: Identify the concept of Inverse Variation

Observe the given equations. An inverse variation or inverse proportion is when the product of two variables is a constant. i.e., xy = k or y = k/x or y = kx. Thus, we can say that if the product of two variables is constant, then they are inversely proportional.

Step 2: Apply the concept to the given equations

Apply this concept to the choices provided. Choice A. \(x=\frac{y}{z}\), which can also be written as \(xz=y\) equivalently, this means the variables x and z have a direct variation with y not inverse. Similarly, Choice B. \(x=\frac{-15z}{y}\) rearranges to \(xy=-15z\) meaning here that x and y directly vary with z, not inversely. For Choice C. \(z=\frac{-15y}{x}\), when rearranged, it gives \(xz=-15y\), again indicating a direct variation between x and z. Only Choice D. \(xz=5y\) adheres to inverse variation as the product of x and z is constant.

Step 3: Identify exceptional case

Among all the equations, only choice D adheres to the inverse variation rule, the rest – Choice A, B, and C do not. The problem asks for the equation that does NOT represent an inverse variation which from our analysis are equations A, B and C. However, we need to choose only one, so any of them could be the answer. Here it's important to note this can be seen as a trick question and the response to this would typically be considered incorrect, as there are three possible valid answers.

Key Concepts

Proportions

A proportion is a relationship between two quantities where they vary together in a predictable way. In mathematics, we often encounter two main types of proportional relationships: direct and inverse proportions.
For direct proportions, as one quantity increases, the other increases at a constant rate too. Imagine

• Doubling the ingredients in a recipe to make double the servings.
• The rate remains constant as the quantities change.

In contrast, inverse proportions involve one quantity increasing while the other decreases, keeping their product constant. Consider a scenario where

• As the speed of a car increases, the time needed to reach a destination decreases.
• The product of speed and time remains constant to cover the same distance.

Therefore, understanding the distinction between direct and inverse proportions helps identify different relationships in equations and everyday scenarios.

Variables

Variables are symbols used in mathematical expressions to represent unknown values. Commonly denoted by letters like \(x\), \(y\), or \(z\), variables allow us to create equations that describe various relationships between quantities.
In an equation such as \(x = \frac{y}{z}\), \(x\), \(y\), and \(z\) are variables. They can take on numerous values, enabling us to explore the relationship between them.
Understanding how variables interact forms the foundation of mathematical modeling and problem-solving.

• Variables can change based on other variables in the equation.
• They help to set up mathematical equations that describe real-world scenarios.

Recognizing how variables relate to each other, in direct or inverse proportions, is crucial for solving equations properly.

Mathematical Equations

Mathematical equations are statements of equality containing variables and constants connected by mathematical operations. They serve as powerful tools to represent and solve problems.
In the context of inverse variation, an equation like \(xy = k\) indicates the inverse relationship, where \(k\) is constant.
This equation can be rewritten to solve for one variable in terms of another, such as \(y = \frac{k}{x}\). This flexibility lets us find solutions based on different known values.

• Equations can model relationships between variables.
• They provide a systematic method for finding unknown values.

The power of mathematical equations lies in their ability to translate complex relationships into manageable formats, allowing for structured analysis and solution finding.