Problem 49 - Algebra 2

Question

Suppose that \(x\) and \(y\) vary inversely. Write a function to model inverse variation. \(x=1.8\) when \(y=-6\)

Step-by-Step Solution

Answer

The function that models the inverse variation between \(x\)and \(y\) is \(xy = -10.8\).

Step 1: Introduce the formula of inverse variation

The equation of inverse variation is given by \(xy = k\), where \(x\) and \(y\) are two quantities that vary inversely and \(k\) is the constant of variation.

Step 2: Substitute the known values into the equation

It is said that \(x = 1.8\) when \(y = -6\). By substituting these values into the equation \(xy = k\), we get \(1.8*(-6) = k, which simplifies to -10.8 = k\). So, the constant of variation is -10.8.

Step 3: Write the equation of inverse variation

Now that we have found the constant of variation (\(k = -10.8\)), we can write the equation of inverse variation as \(xy = -10.8\). This is the function that models the inverse variation between \(x\) and \(y\).

Key Concepts

Algebra

Algebra is like a powerful language we use to describe relationships between numbers and variables. In our scenario, we're dealing with an inverse variation relationship. This is a concept within algebra that shows how two variables are related inversely, meaning as one increases, the other decreases.
In algebra, we often express relationships between quantities using equations. For inverse variation, the equation is written as \(xy = k\), where \(k\) is a constant. This means that the product of the two variables \(x\) and \(y\) will always equal the constant \(k\).
This equation doesn't just show numbers; it lays out a rule for how \(x\) and \(y\) affect each other. It allows us to predict how a change in one variable affects the other. Understanding this is crucial in solving real-world problems using algebraic methods.

Constant of Variation

The constant of variation \(k\) is a very important concept in inverse variation. It's the unchanging value that results from the product of \(x\) and \(y\). This constant helps us form a bridge between the variables, showing how they're connected.
In the given exercise, when we substitute \(x = 1.8\) and \(y = -6\) into the equation \(xy = k\), we find that \(k = -10.8\). This tells us that no matter what \(x\) and \(y\) are, as long as their product equals \(-10.8\), they will always maintain an inverse variation. It's like a fixed rule that \(x\) and \(y\) have to play by.
Knowing \(k\) is also crucial because it allows us to derive one variable if we have the other. This is especially useful in mathematical modeling, where understanding one aspect of a system allows us to determine other aspects based on constant rules.

Functions

Functions are special equations that describe how one quantity changes with another. When dealing with inverse variation, the relationship between the variables can be thought of as a function. This function captures how changing one variable impacts the other in a predictable way.
In the case of inverse variation, the function is expressed as \(y = \frac{k}{x}\). Here, \(y\) is what we get (output) when we substitute a value for \(x\) (input). It's a neat way to encapsulate the relationship in a single expression.
Why use functions? They help us visualize and understand the relationship between variables clearly and concisely. Functions are pivotal in algebra for solving equations, predicting outcomes, and modeling real-world situations accurately.

• They allow for easy computation of outcomes given certain inputs.
• Visual representation using graphs to showcase how variables interact.
• They form the backbone of mathematical modeling and problem-solving.

As you get comfortable with functions, you'll see just how powerful they are in solving a wide range of problems.