Problem 38 - Algebra 1

Question

Decide if the data in the table show direct or inverse variation. Write an equation that relates the variables. $$\begin{array}{|c|c|c|c|c|c|}\hline x & 1 & 3 & 4 & 10 & 0.5 \\\\\hline y & 30 & 10 & 7.5 & 3 & 60 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The data in the table show 'Inverse Variation', and the equation relating the variables is \( y = 30/x \).

Step 1: Define Variation Types

Before starting, it is important to understand two types of variations. One is 'Direct Variation', in which when one quantity increases, the other one too increases, and the ratio of the two is a constant. The second one is 'Inverse Variation', where as one quantity increases, the other one decreases, and their product is a constant.

Step 2: Check for Direct Variation

To check if the variables show direct variation, divide each \( y \) value by its corresponding \( x \) value. Compute as \( 30/1 \), \( 10/3 \), \( 7.5/4 \), \( 3/10 \), \( 60/0.5 \). The outputs are 30, 3.33, 1.875, 0.3, 120. The ratio is not constant, hence this is not a direct variation.

Step 3: Check for Inverse Variation

To check if the variables show inverse variation, multiply each \( y \) value by its corresponding \( x \) value. Compute as \( 30*1 \), \( 10*3 \), \( 7.5*4 \), \( 3*10 \), \( 60*0.5 \). The outputs are 30, 30, 30, 30, 30. The product is constant, hence this is an inverse variation.

Step 4: Write an Equation for Inverse Variation

Since this is the case of inverse variation, the equation would be \( y = k/x \) where \( k \) is the constant of variation. Consider any pair of \( x, y \) to determine the value of \( k \). For example, using \( x = 1 \) and \( y = 30 \), the equation becomes \( 30 = k/1 \) so \( k = 30 \). Hence, the equation that relates the variables for this case is \( y = 30/x \)

Key Concepts

Direct Variation

Direct variation occurs when two variables move in the same direction; that is, as one increases, the other does so as well. In this type of relationship, the ratio of the two variables is always constant. This can be represented mathematically by the equation
\[ y = kx \] where \( k \) is the constant of variation.
For example, if a vehicle travels at a constant speed, the distance traveled directly varies with the time spent traveling.

• If the speed (\( x \)) is directly proportional to the distance (\( y \)), as time (\( x \)) increases, the distance (\( y \)) increases.

To check for a direct variation, you would divide \( y \) values by their corresponding \( x \) values.
If the ratio \( \frac{y}{x} \) remains constant, the variables follow a direct variation.

Constant of Variation

In both direct and inverse variation, the constant of variation plays a crucial role.
It's the number that remains constant when examining the relationship between two variables. For direct variation, as we previously explained, it is represented as \( k \) in the equation \( y = kx \).
For inverse variation, where one variable increases as the other decreases, the constant of variation is also denoted by \( k \). In such cases, the product of the two variables is constant, presented as:
\[ y \times x = k \]In the given exercise, since the product of the variables was consistent at 30, this equation characterizes the inverse variation as \( y = \frac{k}{x} \). Here \( k = 30 \). Calculating \( k \) helps confirm the nature of variation and write an exact relationship between the variables.

Equation Writing

Writing equations that describe variable relationships is essential for understanding their behavior.
In the case of inverse variation, we express the relationship mathematically as:
\[ y = \frac{k}{x} \]Here, \( k \) is the constant of variation.

• By verifying the product \( y \times x \) across different data points for consistency, it became evident that this was an inverse relationship, represented with the constant \( k = 30 \).

To complete the equation writing process, simply include this constant:
\( y = \frac{30}{x} \)This equation now accurately describes how \( y \) changes as \( x \) varies, maintaining the inverse relationship identified by the constant product across all data points. Understanding this equation lets you predict one variable if the other is known, assuming the relationship remains constant.