Problem 68 - Intermediate Algebra

Question

Use the data in each table to find an equation that mathematically describes the relationship between the two quantities. $$ \begin{array}{|c|c|} \hline \text { Seasonal employees } & \text { Employees } \\ \hline 25 & 75 \\ \hline 50 & 100 \\ \hline 60 & 110 \\ \hline 80 & 130 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The relationship is given by the equation \(y = x + 50\), where \(x\) is seasonal employees and \(y\) is total employees.

Step 1: Analyze the Data

Look at the pairs of data given. The first column represents seasonal employees, and the second column represents the total number of employees. Notice that as the number of seasonal employees increases, the total number of employees increases consistently.

Step 2: Identify the Relationship Type

Determine if the relationship between seasonal employees and total employees is linear. Calculate the differences between consecutive seasonal employee values and their corresponding total employee values to see if the increments are consistent.

Step 3: Calculate the Difference

Calculate the increase in the total number of employees for corresponding increases in seasonal employees: - From 25 to 50 seasonal employees, total employees go from 75 to 100 (increase of 25 and 25). - From 50 to 60 seasonal employees, total employees go from 100 to 110 (increase of 10 and 10). - From 60 to 80 seasonal employees, total employees go from 110 to 130 (increase of 20 and 20). The consistent increase indicates a linear relationship with a certain slope.

Step 4: Determine the Equation

Use the data to formulate a linear equation of the form \(y = mx + b\), where \(y\) is total employees, \(x\) is seasonal employees, \(m\) is the slope, and \(b\) is the y-intercept. Calculate the slope \(m = \frac{\text{change in y}}{\text{change in x}} = \frac{25}{25} = 1\). Hence, each seasonal employee corresponds to one additional total employee.

Step 5: Calculate the Y-Intercept

Substitute one of the data points into the equation \(y = mx + b\) to solve for \(b\). Using the first data point (25, 75), \(75 = 1(25) + b\), which simplifies to \(b = 75 - 25 = 50\). So, the equation is \(y = x + 50\).

Step 6: Verify the Equation

Check other data points to verify the accuracy of the equation \(y = x + 50\):- For 50 seasonal employees, \(y = 50 + 50 = 100\).- For 60 seasonal employees, \(y = 60 + 50 = 110\).- For 80 seasonal employees, \(y = 80 + 50 = 130\).The equation holds true for all data points.

Key Concepts

Slope

In the context of a linear equation, the slope is a critical component that defines the steepness and direction of the line. It measures how much the dependent variable (usually represented by \(y\)) changes in response to a change in the independent variable (commonly \(x\)). The formula to calculate the slope \(m\) is given by:\[m = \frac{\text{change in } y}{\text{change in } x}\]In this exercise, we observed a direct linear relationship between the number of seasonal employees (\(x\)) and the total number of employees (\(y\)). By analyzing the data, increases in seasonal employees consistently matched increases in total employees at a ratio of 1:1, meaning that the slope \(m\) is 1. Understanding the slope is essential because:

• A positive slope indicates that as \(x\) increases, \(y\) also increases, showing a direct linear relationship.
• If the slope were negative, it would indicate that \(y\) decreases as \(x\) increases.
• A slope of zero would mean that \(y\) remains constant regardless of the changes in \(x\).

y-intercept

The y-intercept in a linear equation is the point where the line crosses the y-axis. It represents the value of the dependent variable \(y\) when the independent variable \(x\) is zero. Let's look again at the equation derived from the exercise: \[y = mx + b\]Where \(b\) is the y-intercept. Using the provided data, we know that at zero seasonal employees, the total number of employees (\(y\)) would be 50, because when substituting values into the equation:\[75 = 1(25) + b\]Solving for \(b\), we found that:\[b = 75 - 25 = 50\]This calculation shows that even without seasonal employees, there are already 50 total employees. The y-intercept is significant because:

• It helps in understanding the baseline or starting value in a linear relationship.
• In practical terms, it indicates the fixed number of employees who aren't seasonal.
• It gives a full picture of the relationship, as the equation accounts for both variables perfectly.

Linear Relationship

A linear relationship implies that there is a straight-line connection between two variables. In simpler terms, if you graph the relationship, it would appear as a straight line. This concept is at the core of linear equations, where the two components—slope and y-intercept—describe this connection mathematically.From the exercise, you saw that the increase in seasonal employees correlated directly with the increase in total employees. This consistent pattern confirmed that the relationship is linear.Characteristics of a linear relationship include:

• Constant rate of change, which means that for any unit increase in \(x\) there will be a consistent increase or decrease in \(y\).
• Graphically depicted by a straight line that can be represented by the equation \(y = mx + b\).
• Predictability, allowing individuals to reliably use the equation to estimate \(y\) given any \(x\), or vice versa, as shown by the consistent validation through various data points in the problem.

Understanding this helps students see how numerical data translates into a straight line on a graph, a fundamental concept in algebra and further mathematics.