Problem 65 - Understanding Elementary Algebra with Geometry

Question

Suppose that Jane works part-time making deliveries for a caterer. She gets paid a base salary of \(\$ 80\) per day plus \(\$ 15\) for each delivery she makes that day. (a) Letting \(d\) represent the number of deliveries she makes and letting \(A\) represent the amount she earns for each day that she works, write an equation relating \(d\) and \(A\). (b) Sketch the graph of the equation obtained in part (a), representing \(d\) along the horizontal axis. (c) Find the slope of the line graphed in part (b) and relate it to the equation obtained in part (a).

Step-by-Step Solution

Answer

The equation is \(A = 80 + 15d\), the slope is 15, and the graph is a straight line with a y-intercept of 80.

Step 1: Understand the Variables

Determine the variables to be used in the equation. Let \(d\) represent the number of deliveries she makes and \(A\) represent the amount she earns each day.

Step 2: Set Up the Equation

Jane's earnings for each day include a base salary plus additional earnings per delivery. The equation relating \(d\) and \(A\) is \(A = 80 + 15d\).

Step 3: Sketch the Graph

Draw a graph with \(d\) on the horizontal axis and \(A\) on the vertical axis. Plot the y-intercept \(A = 80\) where \(d = 0\), and use the slope to plot additional points.

Step 4: Identify the Slope

The slope of the line is the coefficient of \(d\) in the equation \(A = 80 + 15d\). Here, the slope is \(15\), which indicates that for each additional delivery, Jane earns \$15 more.

Step 5: Relate Slope to the Equation

The slope \(15\) shows the rate at which Jane's earnings increase for each delivery. This slope is consistent across the graph, representing the linear relationship between \(d\) and \(A\).

Key Concepts

Graphing Linear Equations

Graphing linear equations is essential for visualizing mathematical relationships. Here, we consider Jane's earnings, represented by the equation: \( A = 80 + 15d \). In this equation:
Jane's base pay of \(80\) dollars is the y-intercept. This means when she makes zero deliveries, the graph intercepts the y-axis at \(80\).
To graph this, start by plotting the point \(d = 0\) and \(A = 80\).
Next, use the slope (explained below) to find additional points: from the intercept, move up \(15\) units (pay per delivery) and right by \(1\) unit (one delivery). This forms a straight line because the relationship between deliveries and earnings is linear.
Graphing linear equations like this helps understand how variables interact and change together.

Slope-Intercept Form

The slope-intercept form of a linear equation is written as \( y = mx + b \). Here, \(m\) represents the slope, and \(b\) is the y-intercept. In Jane's earnings formula: \( A = 80 + 15d \), we see this form clearly: - \(A\) is like \(y\) (dependent variable, Jane's earnings). - \(d\) is like \(x\) (independent variable, deliveries made). - The number \(15\) is the slope (how much Jane earns per delivery). - The number \(80\) is the y-intercept (base pay per day).
Slope-intercept form makes it easy to identify key information in the equation. Slope shows how quickly earnings change with deliveries, and the intercept shows earnings without deliveries. Understanding this form is crucial for graphing and interpreting linear equations.

Variable Relationship

Linear equations like \( A = 80 + 15d \), represent a clear relationship between two variables. In this case: - \(d\) (number of deliveries) is the independent variable, which Jane controls by making more deliveries. - \(A\) (earnings) is the dependent variable, influenced by the number of deliveries.
For each \(d\) unit increase (one more delivery), \(A\) increases by \(15\) units. This is due to the slope \(15\), indicating the amount earned per delivery.
Understanding variable relationships helps predict how changes in one variable affect the other. Here, we see a direct, proportional relationship: more deliveries lead to higher earnings linearly.
Grasping this relationship is vital for solving real-life problems and interpreting how variables intertwine.