Problem 2 - College Algebra with Corequisite Support

Question

If two variables vary inversely, what will an equation representing their relationship look like?

Step-by-Step Solution

Answer

The equation looks like \(xy = k\), where \(k\) is a constant.

Step 1: Understanding Inverse Variation

When two variables vary inversely, it means that as one variable increases, the other decreases in such a way that the product of the two variables remains constant. This kind of relationship is opposite to direct variation, where one variable increases as the other increases.

Step 2: Identifying Constant Product

If we let the two variables be \(x\) and \(y\), and they vary inversely, then there exists a constant \(k\) such that their product \(xy\) is equal to this constant. Thus, \(xy = k\) is the fundamental equation representing inverse variation.

Step 3: Expressing the Relationship

Given the fundamental relationship \(xy = k\), we can rearrange this to show how one variable depends on the other: \(y = \frac{k}{x}\) or equivalently \(x = \frac{k}{y}\). These forms are derived by solving the equation \(xy = k\) for one of the variables.

Key Concepts

Inverse Relationship

Inverse relationships between variables refer to a situation in which one variable increases while the other decreases in such a way that their product remains constant. This concept is crucial across various scientific and mathematical applications as it models how certain quantities interact with each other. An easy relatable example is the relationship between speed and travel time for a fixed distance. If you increase speed, travel time decreases, and vice versa, while the overall product (distance) remains consistent.In mathematical terms, if two variables, say \(x\) and \(y\), bear such a relationship, they are said to vary inversely. The essence of this inverse relationship is captured by ensuring that multiplying \(x\) with \(y\) always gives a constant product, a mainstay element in this concept.

Constant Product

A constant product is a defining characteristic of an inverse relationship. When two variables vary inversely, they are linked by a constant that governs how they change relative to each other. Simplifying, if you take the product of these two variables, you always arrive at the same fixed value, irrespective of changes in individual variables.Let's denote the variables as \(x\) and \(y\). The product \(xy\) will always equal the constant \(k\). Therefore, the relationship is mathematically represented as:\[ xy = k \]This equation succinctly encapsulates the notion that even if \(x\) increases or decreases, \(y\) adjusts accordingly to maintain the constant product \(k\). This dependability makes it an appealing model for explaining certain real-world scenarios such as pressure and volume of gases in chemistry.

Variation Equations

Variation equations are mathematical expressions that represent how variables relate to each other. In the specific case of inverse variation, the primary equation used is \(xy = k\). This equation is versatile and can be rearranged depending on which variable you have and which you want to find.For instance, if we know the value of \(x\) and the constant \(k\), then:\[ y = \frac{k}{x} \]Similarly, if \(y\) is known:\[ x = \frac{k}{y} \]These forms of variation equations help determine one variable as long as both \(k\) and the other variable are known. They bring clarity to the often complex interactions between variables by simplifying the relationships into solvable queries, which is particularly useful in fields like physics, engineering, and economics.