Question

Claire goes to a used bookstore in search of novels and biographies, She spends a total of \(\$ 24\), buying \(b\) biographies for \(\$ 3\) each and \(n\) novels for \(\$ 4\) each. Write an equation relating the numbers of each type of book Claire bought. Graph your equation, putting \(b\) on the horizontal axis.

Short answer

The equation is \[3b + 4n = 24\].

Step-by-step solution

Identify the total spent

Claire spends a total of \(\$24\). This is the sum of the amounts she spends on biographies and novels.

Calculate the cost of biographies

Claire buys biographies for \(\(3\) each. Let the number of biographies be \(b\). The total cost of the biographies is \(\)3b\).

Calculate the cost of novels

Claire buys novels for \(\(4\) each. Let the number of novels be \(n\). The total cost of the novels is \(\)4n\).

Write the equation

Since the total expenditure is \($24\), the equation relating the amounts spent can be written as: \[3b + 4n = 24\]

Rearrange the equation

To graph this equation, solve for \(n\) in terms of \(b\): \[4n = 24 - 3b\] Divide both sides by 4: \[n = \frac{24 - 3b}{4}\]

Create a table of values

Choose values for \(b\) and calculate corresponding \(n\) values. For example: if \(b = 0\), then \[n = \frac{24 - 3(0)}{4} = 6\] if \(b = 2\), then \[n = \frac{24 - 3(2)}{4} = 4.5\] if \(b = 4\), then \[n = \frac{24 - 3(4)}{4} = 3\]

Plot the points and graph the equation

On a coordinate plane, plot the points found in the table. Connect the points to form a straight line. The horizontal axis represents \(b\), and the vertical axis represents \(n\).

Key concepts

Cost Calculation

When dealing with linear equations involving costs, it is crucial to understand each component of the total cost. For Claire's problem, we are given the total amount spent and the individual prices of biographies and novels.
To start with, identify the total expenditure, which is \$24. Next, calculate the total cost for each type of book:

• The cost of biographies: Claire buys ´\(b\)´ biographies for \$3 each, which is represented as ´\(3b\)´.
• The cost of novels: She buys ´\(n\)´ novels for \$4 each, represented as ´\(4n\)´.

Now, combine these to set up the equation for the total cost:

\(3b + 4n = 24\)

This equation shows that the sum of the money spent on biographies and novels totals \$24.

Graphing Linear Equations

Graphing linear equations helps visualize the relationship between two variables. For Claire's book purchases, you need to graph ´\(3b + 4n = 24\)´ on a coordinate plane.
First, solve the equation for one variable in terms of the other. Here, we'll express ´\(n\)´ in terms of ´\(b\)´:
\[4n = 24 - 3b\]
Divide both sides by 4 to isolate ´\(n\)´:
\[n = \frac{24 - 3b}{4}\]
.Next, create a table of values for ´\(b\)´ and determine corresponding ´\(n\)´ values. Pick easy values for ´\(b\)´, like 0, 2, and 4, and calculate:

• If ´\(b = 0\)´, then ´\(n = 6\)´.
• If ´\(b = 2\)´, then ´\(n = 4.5\)´.
• If ´\(b = 4\)´, then ´\(n = 3\)´.

Plot these points on a graph, with ´\(b\)´ on the horizontal axis and ´\(n\)´ on the vertical axis.
Finally, draw a straight line through the points to represent the equation.

Solving for a Variable

Solving for a variable means expressing one variable in terms of another. This is a critical step in understanding and graphing linear equations. Let's break it down using Claire's book example.
You start with the equation:
\[3b + 4n = 24\]
To solve for ´\(n\)´, follow these steps:

• Isolate the term involving ´\(n\)´: ´\(4n = 24 - 3b\)´.
• Then, divide every term by 4 to solve for ´\(n\)´:
  \[n = \frac{24 - 3b}{4}\]

By solving for ´\(n\)´, we can easily generate points for graphing. Additionally, it helps in understanding how one variable changes with respect to the other. This is the foundation of interpreting linear relationships in equations.
When you grasp this, it becomes simpler to see how changing the number of biographies affects the number of novels Claire can buy, all within her budget of \$24.