Problem 26 - Algebra 1

Question

The variables x and y vary inversely. Use the given values to write an equation that relates x and y. $$x=\frac{13}{5}, y=5$$

Step-by-Step Solution

Answer

The inverse variation equation that represents the relationship between x and y is \(xy = 13\).

Step 1: Understanding Inverse Variation

If two variables x and y vary inversely, this means that their product is a constant. This can be written in mathematical form as \(xy = k\), where k is the constant of variation.

Step 2: Calculating the Constant of Variation

Substitute the given x and y values into the formula. Therefore, when \(x= \frac{13}{5}\) and \(y=5\), we can calculate \(k = x * y = \frac{13}{5} * 5 = \frac{65}{5} = 13\). Here, 13 is the constant of variation.

Step 3: Writing the Inverse Variation Equation

Now we can put the constant k into the original formula \(xy = k\), we have the inverse variation equation \(xy = 13\). This equation represents the inverse relationship between x and y.

Key Concepts

Constant of Variation

In inverse variation, the constant of variation is a key concept. It defines how two variables, such as \(x\) and \(y\), are related to each other inversely. When two variables vary inversely, it means that as one increases, the other decreases in such a way that their product remains constant.
This constant is represented by \(k\) in the formula \(xy = k\).

• The constant \(k\) does not change as \(x\) and \(y\) change.
• Understanding this constant is crucial to grasping the idea of inverse variation.
• Once you know the constant of variation, you can easily predict how changes in one variable affect the other.

Calculating \(k\) is straightforward; simply multiply the given values of \(x\) and \(y\) to find their product.

Variables

Variables are symbols used to represent numbers or values in mathematical expressions and equations. In the context of inverse variation, \(x\) and \(y\) are the variables involved. These variables can change, unlike constants, which stay the same.
Variables in inverse variation have an interesting characteristic:

• As one variable increases, the other decreases to maintain the constant product.
• This relationship can model real-world situations, like speed and time taken for a journey at constant distance.
• Being comfortable with manipulating variables is essential for solving inverse variation problems.

By grasping how variables interact in equations, you'll develop a deeper understanding of algebra.

Product of Variables

The product of variables \(x\) and \(y\) in inverse variation is pivotal because it always equals the constant of variation (\(k\)). This relationship is described by the equation \(xy = k\).
Let's break this down further:

• Multiply \(x\) and \(y\) to find their product.
• This product holds steady, even if values of \(x\) and \(y\) change, emphasizing the inverse nature.
• The equation demonstrates the interplay of \(x\) and \(y\), showcasing how they balance each other out.

Remember, the wisdom of inverse variation lies in grasping how this product remains unchanged.

Mathematical Equation

A mathematical equation is a statement that shows the equality of two expressions. In terms of inverse variation, this equation relates the variables \(x\) and \(y\) by equating their product to a constant. The equation \(xy = k\) is an example of such an equation.
Some key insights include:

• The equation serves as a tool to describe and predict the relationship between \(x\) and \(y\).
• Writing an equation allows you to solve for one variable when the other is known.
• Equations are fundamental to expressing mathematical relationships and solving problems.

Mastering how to write and manipulate equations is a valuable skill in mathematics, and it all starts with understanding their structure and meaning.