Problem 15 - Algebra A Combined Function

Question

If \(y\) varies inversely as \(x\), find the constant of variation and the inverse variation equation for each situation. \(y=100\) when \(x=7\)

Step-by-Step Solution

Answer

The constant of variation is 700, and the equation is \( y = \frac{700}{x} \).

Step 1: Understand Inverse Variation

In inverse variation, the product of the two variables is constant. This means that if \( y \) varies inversely as \( x \), then \( y \cdot x = k \), where \( k \) is the constant of variation.

Step 2: Substitute Known Values

We are given that \( y = 100 \) when \( x = 7 \). Using the formula for inverse variation, substitute the known values into \( y \cdot x = k \) to find \( k \). Thus, \( 100 \cdot 7 = k \).

Step 3: Calculate the Constant of Variation

Calculate \( k \) by multiplying the values from Step 2: \( 100 \cdot 7 = 700 \). So, the constant of variation \( k = 700 \).

Step 4: Write the Inverse Variation Equation

With the constant \( k = 700 \), the inverse variation equation can be written as \( y = \frac{k}{x} \). Substitute \( k \) to get \( y = \frac{700}{x} \).

Key Concepts

Constant of Variation

The constant of variation is a key concept in inverse variation problems. It represents a fixed number that remains unchanged even as the values of the variables might fluctuate. In the case of inverse variation, where one value increases while the other decreases, the product remains the same. This constant is denoted by the symbol, usually represented by the letter \( k \). To find the constant of variation, you simply need to multiply the values of the two variables involved. For example, if \( y \) is 100 when \( x \) is 7 in an inverse variation scenario, the constant of variation \( k \) is calculated as:

• Multiply \( y \) by \( x \), so: \( 100 \times 7 \).
• This gives you \( k = 700 \).

This \( k \) value is essential for writing the inverse variation equation and understanding the relationship between the variables.

Inverse Variation Equation

The inverse variation equation is a mathematical expression that shows how two variables relate inversely. When one variable increases, the other decreases, but their product is constant. If \( y \) varies inversely as \( x \), the relationship can be represented by:

• \( y \cdot x = k \)
• Or, reformulated, as \( y = \frac{k}{x} \)

For the example where \( y = 100 \) and \( x = 7 \), we found the constant \( k \) to be 700. Substituting \( k \) into the inverse variation formula gives:

• \( y = \frac{700}{x} \)

This equation allows you to determine the value of \( y \) for any given \( x \) and is essential for solving problems where the relationship between the variables is inversely proportional.

Multiplying Variables

In the context of inverse variation, multiplying variables is how you find the constant of variation. For inverse relationships, the essential principle is that as one variable increases, the product of the two variables remains consistent. Let's break down the process:

• Identify the variables: Initially ascertain which values you have. For instance, \( y = 100 \) and \( x = 7 \).
• Calculate the product: Multiply the known values to find the constant, \( 100 \times 7 = 700 \).

This simple multiplication is fundamental to understanding how the variables interact and remain in balance according to the rule of inverse variation. Keeping the concept of multiplying variables in mind helps grasp why the inverse variation equation maintains a steady product, even as individual variable values change.