Question

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.

Answer 1

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.

Answer 2

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

Answer 3

Let \( b \) represent the number of board books Santiago buys, and let \( s \) represent the number of soft cover books he buys.

According to the prices given in the problem:
- Each board book costs $1.50.
- Each soft cover book costs $0.50.

Santiago spends a total of $20, so we can write the equation representing this situation as:

\[
1.5b + 0.5s = 20
\]

This equation indicates that the total cost of the board books and soft cover books equals $20.

Answer 4

To check if Santiago could have bought 10 board books and 5 soft cover books, we can plug these values into the equation we derived earlier:

\[
1.5b + 0.5s = 20
\]

Substituting \( b = 10 \) (the number of board books) and \( s = 5 \) (the number of soft cover books):

\[
1.5(10) + 0.5(5) = 20
\]

Calculating each term:

\[
1.5 \times 10 = 15
\]
\[
0.5 \times 5 = 2.5
\]

Now, adding these results:

\[
15 + 2.5 = 17.5
\]

Since \( 17.5 \neq 20 \), this means that the total cost of 10 board books and 5 soft cover books is $17.50, which does not equal the $20 that Santiago spent.

**Conclusion:** It is not possible for Santiago to have bought 10 board books and 5 soft cover books, as this combination does not satisfy the equation he needed to meet the total spending of $20.

Answer 5

To graph the equation \( 1.5b + 0.5s = 20 \), we can first rearrange it into slope-intercept form or find points that satisfy the equation.

### Step 1: Rearranging the Equation

We can rearrange the equation to express \( s \) in terms of \( b \):

\[
0.5s = 20 - 1.5b
\]

Now, multiply everything by 2 to eliminate the decimal:

\[
s = 40 - 3b
\]

### Step 2: Identifying Points

We can find a few values of \( b \) and the corresponding \( s \):

1. **When \( b = 0 \)**:
\[
s = 40 - 3(0) = 40 \quad \Rightarrow \quad (0, 40)
\]

2. **When \( b = 10 \)**:
\[
s = 40 - 3(10) = 40 - 30 = 10 \quad \Rightarrow \quad (10, 10)
\]

3. **When \( b = 20 \)**:
\[
s = 40 - 3(20) = 40 - 60 = -20 \quad \Rightarrow \quad (20, -20) \text{ (not valid since s can't be negative)}
\]

4. **When \( b = 5 \)**:
\[
s = 40 - 3(5) = 40 - 15 = 25 \quad \Rightarrow \quad (5, 25)
\]

So we have the valid points:
1. \( (0, 40) \)
2. \( (5, 25) \)
3. \( (10, 10) \)

### Step 3: Graphing the Points

1. Plot the points on graph paper:
- Point \( (0, 40) \)
- Point \( (5, 25) \)
- Point \( (10, 10) \)

2. Draw a line through these points. This line represents the equation \( 1.5b + 0.5s = 20 \).

### Step 4: Graphing Notes

- The x-axis represents the number of board books (\( b \)).
- The y-axis represents the number of soft cover books (\( s \)).
- The line will be a straight line with a negative slope, showing the trade-off between the number of board books and soft cover books Santiago could buy.

### Important Consideration

Since neither board books nor soft cover books can be negative in this context, the graph should only be considered in the first quadrant (\( b \geq 0 \) and \( s \geq 0 \)).

Feel free to plot points and draw the line on your graph paper based on the calculations above.