Problem 26 - Algebra 1: Concepts and Skills

Question

The variables \(x\) and \(y\) vary inversely. Use the given values to write an equation that relates \(x\) and \(y .\) $$ x=10.5, y=7 $$

Step-by-Step Solution

Answer

The inverse variation equation that relates \(x\) and \(y\) is \(xy = 73.5\).

Step 1: Understand Inverse Variation

In an equation, an inverse variation means product of the two variables is constant, that is \(xy = k\).

Step 2: Insert Given Values Into Equation

Insert the given values into the inverse variation equation. That is, plugging in \(10.5\) for \(x\) and \(7\) for \(y\) in the equation \(xy = k\). Thus \(10.5 \times 7 = k\). Then, calculate the product.

Step 3: Determine the constant of variation

By calculating the product of \(10.5\) and \(7\), we get \(73.5\). Therefore, \(k = 73.5\).

Step 4: Write the equation that relates \(x\) and \(y\)

Express the equation by replacing \(k\) with \(73.5\). Therefore, the inverse variation equation that relates \(x\) and \(y\) is \(xy = 73.5\).

Key Concepts

Algebraic Equation

An algebraic equation is a statement of equality between two algebraic expressions. It includes variables, numbers, and operation symbols such as plus, minus, multiplication, and division. The primary aim of these equations is to find unknown values that make the equation true.

For example, the equation relating the variables in our inverse variation problem, represented as an algebraic equation, is initially formed as \(xy = k\), where \(k\) is the constant of variation and \(x\) and \(y\) are the variables. To understand and find a solution, we replace the variables with the given numbers and solve for \(k\), resulting in an explicit equation that can be used for further computation.

Constant of Variation

The constant of variation, denoted as \(k\), is a key concept in direct and inverse variation problems. In the context of an inverse variation, \(k\) represents the constant product of the two variables involved. In other words, regardless of the values that \(x\) and \(y\) might take, their product will always equal \(k\).

This constant is unique to the relationship between the variables and must be determined from given values. As observed in our problem, inserting \(x=10.5\) and \(y=7\) into \(xy = k\), we calculate \(k\) to be 73.5, which becomes an essential part of the algebraic equation for this set of variables.

Inverse Proportionality

Inverse proportionality, also known as inverse variation, is a concept where one value increases as the other decreases. In mathematical terms, two variables are inversely proportional if their product is a constant \(k\). The formula \(xy = k\) represents this type of relationship.

In our textbook problem, \(x\) and \(y\) are inversely proportional because as \(x\) increases, \(y\) must decrease to maintain the constant product \(k\), and vice versa. In practical terms, this means that in any set of inverse proportional quantities, a higher number of one will balance out a correspondingly lower number of the other to keep the product the same. This concept is fundamental across various scientific principles and real-world applications.