Santiago is working at his mom’s day care this summer. His mom has asked him to check out the library book sale to look for books they might be able to use in the day care. When he gets to the book sale, he sees that board books are $1.50 and soft cover books are $0.50. He picks out a variety of books and spends $20.

Question

Write a two-variable equation to represent this situation. Be sure to define the variables you use in your equation.
Is it possible that Santiago bought 10 board books and five soft cover books? Use your equation from part A to check. Explain your answer.
Graph the solution set for the equation you wrote in part A on a piece of graph paper. Note: At this point, do not yet consider any constraints given by the situation.
What are the constraints of the situation? Describe them.
Use your graph to help you make a list of all viable solutions to the problem.
Suppose Santiago bought 34 books. Can you figure out how many of each type of book he bought? Explain.

Answer 1

Let's break down the task into manageable parts.

Part A: Write a Two-Variable Equation

Define the variables:

• Let \( x \) be the number of board books Santiago buys.
• Let \( y \) be the number of softcover books Santiago buys.

The cost of board books is $1.50 each, and the cost of softcover books is $0.50 each. Since Santiago spent a total of $20, we can create the following equation representing the situation:

\[ 1.5x + 0.5y = 20 \]

Part B: Check if Santiago bought 10 board books and 5 softcover books

Substitute \( x = 10 \) and \( y = 5 \) into the equation:

\[ 1.5(10) + 0.5(5) = 15 + 2.5 = 17.5 \]

Since \( 17.5 \) is not equal to \( 20 \), it is not possible that Santiago bought 10 board books and 5 softcover books.

Part C: Graph the Solution Set

To graph the equation \( 1.5x + 0.5y = 20 \), we can rearrange it into slope-intercept form:

First, isolate \( y \):

\[ 0.5y = 20 - 1.5x \] \[ y = 40 - 3x \]

Now you can plot the line \( y = 40 - 3x \).

Find intercepts:

• When \( x = 0 \): \[ y = 40 \quad (0, 40) \]

• When \( y = 0 \): \[ 0 = 40 - 3x \implies 3x = 40 \implies x = \frac{40}{3} \approx 13.33 \quad \left(\frac{40}{3}, 0\right) \]

Part D: Constraints of the Situation

The constraints for this situation are:

1. \( x \geq 0 \): Santiago cannot buy a negative number of board books.
2. \( y \geq 0 \): Santiago cannot buy a negative number of softcover books.
3. \( 1.5x + 0.5y = 20 \): The total money spent must equal $20.

Part E: List of All Viable Solutions

To find viable pairs \( (x, y) \), we can solve for \( y \) in terms of \( x \) and impose the constraints:

1. Rearranging gives \( y = 40 - 3x \). For \( y \) to be non-negative:

   \[ 40 - 3x \geq 0 \implies x \leq \frac{40}{3} \approx 13.33 \]

2. Since \( x \) must also be a whole number, \( x \) can take values from \( 0 \) to \( 13 \).

Then we calculate \( y \) for each non-negative integer \( x \):

• \( x = 0 \): \( y = 40 \) → (0, 40)
• \( x = 1 \): \( y = 37 \) → (1, 37)
• \( x = 2 \): \( y = 34 \) → (2, 34)
• ...
• \( x = 13 \): \( y = 1 \) → (13, 1)

Viable pairs include \( (0, 40), (1, 37), (2, 34), \dots, (13, 1) \).

Part F: Suppose Santiago bought 34 books

If Santiago bought a total of 34 books, we now have a second equation:

\[ x + y = 34 \]

Now we have a system of equations to solve:

1. \( 1.5x + 0.5y = 20 \)
2. \( x + y = 34 \)

From the second equation, we can express \( y \) in terms of \( x \):

\[ y = 34 - x \]

Substituting this into the first equation:

\[ 1.5x + 0.5(34 - x) = 20 \] \[ 1.5x + 17 - 0.5x = 20 \] \[ 1.0x + 17 = 20 \] \[ x = 3 \]

Now substituting \( x \) back to find \( y \):

\[ y = 34 - 3 = 31 \]

Therefore, Santiago bought 3 board books and 31 softcover books.

Conclusion

This structured approach shows how to define equations based on a real-world scenario, verify data points, graph solutions, establish constraints, and solve a system of equations effectively.