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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.

A. Write a two-variable equation to represent this situation. Be sure to define the variables you use in your equation.
B. 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.
C. 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.
D. What are the constraints of the situation? Describe them.
E. Use your graph to help you make a list of all viable solutions to the problem.
F. Suppose Santiago bought 34 books. Can you figure out how many of each type of book he bought? Explain.

2 answers

Let's break down the problem step-by-step:

A. Two-Variable Equation

  1. Define the variables:

    • Let \( b \) = the number of board books.
    • Let \( s \) = the number of soft cover books.

  2. Set up the equation: The cost of board books is $1.50 each, and the cost of soft cover books is $0.50 each. Santiago spends a total of $20. Hence, the equation representing this situation is:

    \[ 1.5b + 0.5s = 20 \]

B. Checking for 10 Board Books and 5 Soft Cover Books

  • Substitute \( b = 10 \) and \( s = 5 \) into the equation:

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

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

C. Graph the Solution Set

To graph the equation \( 1.5b + 0.5s = 20 \):

  1. Convert it to slope-intercept form (solving for \( s \)):

    \[ 0.5s = 20 - 1.5b \] \[ s = 40 - 3b \]

  2. Identify two points to graph the line:

    • When \( b = 0 \): \( s = 40 \) (point (0, 40))
    • When \( s = 0 \): \( 1.5b = 20 \) → \( b = \frac{20}{1.5} \approx 13.33 \) (point (13.33, 0))

  3. Plot the points and draw the line.

D. Constraints of the Situation

  1. Non-negativity constraint: Santiago cannot buy a negative number of books. Therefore:

    • \( b \geq 0 \)
    • \( s \geq 0 \)

  2. Integer constraint: Since books are discrete items, both \( b \) and \( s \) must be non-negative integers.

E. List of All Viable Solutions

To find viable integer solutions, we can use the equation \( 1.5b + 0.5s = 20 \) and rearrange to solve for \( s \):

Using \( s = 40 - 3b \), we can find integers:

  • If \( b = 0 \): \( s = 40 \)
  • If \( b = 1 \): \( s = 37 \)
  • If \( b = 2 \): \( s = 34 \)
  • If \( b = 3 \): \( s = 31 \)
  • If \( b = 4 \): \( s = 28 \)
  • If \( b = 5 \): \( s = 25 \)
  • If \( b = 6 \): \( s = 22 \)
  • If \( b = 7 \): \( s = 19 \)
  • If \( b = 8 \): \( s = 16 \)
  • If \( b = 9 \): \( s = 13 \)
  • If \( b = 10 \): \( s = 10 \)
  • If \( b = 11 \): \( s = 7 \)
  • If \( b = 12 \): \( s = 4 \)
  • If \( b = 13 \): \( s = 1 \)

Integer solutions are \( (b, s) \): (0, 40), (1, 37), (2, 34), (3, 31), (4, 28), (5, 25), (6, 22), (7, 19), (8, 16), (9, 13), (10, 10), (11, 7), (12, 4), (13, 1).

F. If Santiago Bought 34 Books

To find how many of each type of book Santiago bought and knowing he bought a total of 34 books, we set up a second equation based on the total number of books:

\[ b + s = 34 \]

Now we have two equations:

  1. \( 1.5b + 0.5s = 20 \)
  2. \( b + s = 34 \)

From the second equation, we can express \( s \): \[ s = 34 - b \]

Substituting \( s \) into the first equation gives:

\[ 1.5b + 0.5(34 - b) = 20 \] \[ 1.5b + 17 - 0.5b = 20 \] \[ 1b + 17 = 20 \implies b = 3 \]

Now substitute \( b \) back to find \( s \): \[ s = 34 - 3 = 31 \]

So, Santiago bought 3 board books and 31 soft cover books.

Can you simplify the answer for f