Problem 31 - Algebra 1

Question

The variables x and y vary inversely. Use the given values to write an equation that relates x and y. $$x=45, y=\frac{3}{5}$$

Step-by-Step Solution

Answer

The equation that relates x and y for the given inversely varying values is \(y = 27/x\).

Step 1: Understand the concept of inverse variation

The variables x and y vary inversely, which means the product of x and y is constant: \(xy = k\), where k is a constant.

Step 2: Find the value of k

Substitute the given values of x and y into the equation to find the value of k.\nLet's substitute \(x=45\) and \(y=3/5\) into the equation: \(45*(3/5) = k\), so \(k = 27\).

Step 3: Write the equation that relates x and y

The equation that relates x and y, for this case of inverse variation, will be: \(xy = k\), or alternatively \(y = k/x\). Substituting k value, got the equation: \(y = 27/x\).

Key Concepts

Direct and Inverse Variation

Understanding the core principles behind direct and inverse variation is foundational in algebra. These concepts explain how two variables relate to each other in a proportional relationship, either increasing or decreasing together.

Direct variation occurs when two variables change at the same rate, meaning as one variable increases, the other also increases. The equation representing a direct variation is of the form \( y = kx \), where \( k \) represents the constant of proportionality. On the other hand, inverse variation represents a situation where one variable increases as the other decreases at a consistent rate. The equation for inverse variation is \( xy = k \) or \( y = \frac{k}{x} \), showing the multiplicative relationship between the variables that keeps the product, represented by \( k \), constant.

Writing Equations

When faced with the task of writing equations, it's crucial to translate the given relationship between variables into mathematical terms. This process involves identifying whether the variation is direct or inverse and applying the correct formulaic structure.

With inverse variation, the key is to set up an equation where the product of the two variables equals a constant. To perform this task effectively, first identify the values given for the variables and then determine the constant of proportionality by multiplying them together. Once you have the constant, you can construct the equation that models the relationship for all values of the variables.

Constant of Proportionality

The constant of proportionality is the unchanging value that relates the variables in both direct and inverse variation equations. In the context of inverse variation, this constant is especially pivotal because it anchors the varying quantities to a predictable model.

To find this constant, simply multiply the known variable values at one point in their relationship. For instance, with \( x = 45 \) and \( y = \frac{3}{5} \), you calculate the constant \( k \) by multiplying these two numbers, yielding \( k = 27 \). This constant allows you to understand and predict the values of one variable based on the other throughout their inverse relationship.

Algebraic Expressions

Algebraic expressions serve as the language through which we describe patterns and relationships between quantities. They are comprised of variables, numbers, and operations that come together to model these relationships.

In inverse variation problems, the algebraic expression that models the relationship takes a specific form that denotes the constant product of the variables involved. Simplifying and manipulating such expressions are standard algebraic skills that are crucial for understanding and solving variation problems. The elegance of these expressions is that they encapsulate complex relationships in a single statement, such as \( y = \frac{k}{x} \), which you can analyze and interpret in terms of the variables involved.