Problem 23 - 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 \(y .\) $$ x=5, y=\frac{13}{5} $$

Step-by-Step Solution

Answer

The equation that relates \(x\) and \(y\) is \(xy = 13\).

Step 1: Calculate the Constant of Variation

First, substitute the given values of \(x\) and \(y\) into the inverse variation equation \(xy = k\) to find the value of \(k\). Using \(x = 5\) and \(y = 13/5\), the equation becomes \(5 * 13/5 = k\). Therefore, \(k = 13\).

Step 2: Write the Inverse Variation Equation

Now, write the equation that relates \(x\) and \(y\) using the value of \(k\) found in the previous step. The inverse variation equation is \(xy = k\), so substituting \(k = 13\), we get \(xy = 13\).

Key Concepts

Constant of Variation

Understanding the constant of variation is crucial when dealing with inverse variation equations. In these equations, the constant of variation (\( k \)) is a number that remains the same as the variables change proportionately in the inverse manner. This means that as one variable increases, the other decreases at a rate such that their product always equals the constant of variation. For example, in the equation \( xy = k \) given in the problem, the product of \( x \) and \( y \) will always be the same value, which is \( k \).

To find the constant of variation, you simply multiply the known values of \( x \) and \( y \) together. Let’s look at the provided exercise: since \( x = 5 \) and \( y = 13/5 \) are given, you multiply them to get \( k \). Therefore, \( k = 5 \times \frac{13}{5} = 13 \). This value of \( k \) reflects the specific, unchanging relationship between \( x \) and \( y \) for this particular inverse variation.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions. They often include variables and constants and can take various forms, such as linear, quadratic, or, as in our case, inverse variation equations. In an algebraic equation like the inverse variation equation \( xy = k \), the goal is to manipulate the equation to solve for one variable in terms of the other, while respecting the constant of variation.When working with algebraic equations, it's important to remember basic algebra principles such as performing the same operation on both sides of the equation to maintain equality. Additionally, understanding how to isolate a variable, combine like terms, and use distribution and factoring can all be necessary to find a solution. In this exercise, the ease of the algebra involved belies the importance of understanding the underlying principles in order to correctly form and solve more complex equations in the future.

Proportional Relationships

In mathematics, a proportional relationship exists between two quantities when they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently, if they maintain a constant ratio. Inverse variation is a type of proportional relationship where the product of the two variables is always the same constant, as opposed to direct variation wherein the ratio of the two variables is constant.

In the context of the example \( xy = k \) given in the problem, \( x \) and \( y \) share an inverse proportional relationship because as \( x \) increases, \( y \) decreases in such a way that the product \( xy \) remains constant at \( k \). This inverse relationship can be a little less intuitive than a direct proportional relationship, where an increase in one variable leads to a proportional increase in the other. Nevertheless, understanding inverse variation is key for analyzing many real-world phenomena, such as the intensity of light as it moves away from the source, or the pressure and volume relationship in gases known as Boyle's law.