Problem 11 - Elementary Algebra

Question

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=2, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=4\).

Step-by-Step Solution

Answer

a. \( k = 12 \) b. \( y = \frac{12}{x} \) c. \( y = 3 \)

Step 1: Understand Inverse Variation

In an inverse variation, the product of two variables is equal to a constant. This means if one variable increases, the other decreases. The relationship is given by the formula: \[ x \times y = k \] where \( k \) is the constant of proportionality.

Step 2: Find the Constant of Proportionality

Use the given values \( x = 2 \) and \( y = 6 \) to determine \( k \). \[ 2 \times 6 = k \] Thus, \( k = 12 \).

Step 3: Write the Equation for the Inverse Variation

Using the constant of proportionality, the equation representing the inverse variation: \[ x \times y = 12 \] or equivalently, \[ y = \frac{12}{x} \]

Step 4: Find \( y \) When \( x = 4 \)

Substitute \( x = 4 \) into the equation \[ y = \frac{12}{4} \] Simplifying: \[ y = 3 \]

Key Concepts

Understanding the Constant of Proportionality

In inverse variation, the product of two variables is always equal to a constant value. This constant is known as the 'constant of proportionality.' It helps in defining the specific relationship between the variables involved. Here, we know that when one variable increases, the other one decreases in such a way that their product remains constant.
To find this constant, you use the given values from the problem. For instance, if you are provided that when \(x = 2\) and \(y = 6\), the constant of proportionality \(k\) can be found as:
\[k = x \times y \]
Given \(x = 2\) and \(y = 6\), we calculate:
\[k = 2 \times 6 = 12 \]
So, the constant of proportionality, \(k\), is 12.

Formulating the Inverse Variation Equation

Once you know the constant of proportionality, you can write the inverse variation equation. An inverse variation equation expresses the relationship between two variables such that their product is equal to the constant.
The general form of an inverse variation equation is:
\[ x \times y = k \]
Using our previously found constant of proportionality \(k = 12\), we can write:
\[ x \times y = 12 \]
This equation tells us that no matter what values \(x\) and \(y\) take, their product will always be 12.
To make it easier, we can solve for \(y\):
\[ y = \frac{12}{x} \]
This form shows how \(y\) varies inversely as \(x\) changes.

Solving Inverse Variation Problems

Solving inverse variation problems involves using the inverse variation equation to find unknown variable values.
Consider an example where we need to find \(y\) when \(x = 4\). First, we use the inverse variation equation:
\[ y = \frac{12}{x} \]
Next substitute \(x = 4\):
\[ y = \frac{12}{4} \]
Simplifying, we get:
\[ y = 3 \]
Therefore, when \(x = 4\), \(y = 3\). By understanding how to substitute values into the equation, you can solve various inverse variation problems effortlessly.