Santiago is working at his mom’s day care this summer. His mom has asked him to check out the

Question

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.

Write a two-variable equation to represent this situation. Be sure to define the variables
you use in your equation.
variety of books and spends $20.

Answer 1

To represent the situation described, we can define two variables:

Let:

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

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 on these books.

The equation representing this situation will be:

\[ 1.50b + 0.50s = 20 \]

This equation states that the total cost of the board books (1.50 times the number of board books) plus the total cost of the soft cover books (0.50 times the number of soft cover books) equals $20.

Answer 2

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.

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 3

We can use the previously established equation to check whether Santiago could have bought 10 board books and 5 soft cover books.

Recall the equation:

\[ 1.50b + 0.50s = 20 \]

We will substitute \( b = 10 \) (the number of board books) and \( s = 5 \) (the number of soft cover books) into the equation:

\[ 1.50(10) + 0.50(5) \]

Calculating each term:

\[ 1.50 \times 10 = 15 \] \[ 0.50 \times 5 = 2.5 \]

Now, we add these two amounts together:

\[ 15 + 2.5 = 17.5 \]

Since \( 17.5 \) is not equal to \( 20 \), Santiago could not have bought 10 board books and 5 soft cover books using $20. Therefore, this combination does not satisfy the equation.

Conclusion: Santiago could not have purchased 10 board books and 5 soft cover books because the total cost would only be $17.50, which is less than the $20 he spent.

Answer 4

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.

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 5

While I cannot physically graph the equation for you, I can guide you through the steps to graph the solution set of the equation \( 1.50b + 0.50s = 20 \).

Steps to Graph the Equation:

1. Rewrite the Equation in a Straight-Line Format: We can express \( s \) in terms of \( b \): \[ 0.50s = 20 - 1.50b \] \[ s = \frac{20 - 1.50b}{0.50} \] Simplifying further: \[ s = 40 - 3b \]

2. Find Intercepts:

   • \( b \)-Intercept (where \( s = 0 \)): \[ 0 = 40 - 3b \implies 3b = 40 \implies b = \frac{40}{3} \approx 13.33 \]

   • \( s \)-Intercept (where \( b = 0 \)): \[ s = 40 - 3(0) \implies s = 40 \]

3. Plot the Points:

   • \( (0, 40) \): This is the \( s \)-intercept.
   • \( \left(\frac{40}{3}, 0\right) \) or approximately \( (13.33, 0) \): This is the \( b \)-intercept.

4. Draw the Line:

   • Use a ruler to connect the two intercepts. This line represents all the combinations of board books \( b \) and soft cover books \( s \) that can be purchased for $20.

5. Label the Axes:

   • Label the x-axis as "Number of Board Books (b)".
   • Label the y-axis as "Number of Soft Cover Books (s)".
   • Mark the important points like the intercepts on the axes.

Graph Behavior:

• The line will slope downward from left to right because as Santiago buys more board books, he can afford fewer soft cover books with the same amount of money.

Conclusion:

You now have a line on your graph that represents the equation \( 1.50b + 0.50s = 20 \). Remember, the solution set includes all the points on this line, meaning any combination of board books and soft cover books that equals $20 in total cost.

If you have constraints related to the problem (e.g., only whole numbers of books), you can use the line as a reference for further filtering valid solutions.

Answer 6

where should the lines be

Answer 7

To graph the equation \(1.50b + 0.50s = 20\) on a Cartesian plane with the variable definitions:

• x-axis: Number of Board Books \(b\)
• y-axis: Number of Soft Cover Books \(s\)

Here’s how you can interpret where the line should be positioned:

Steps to Graph the Line

1. Identify the x-intercept and y-intercept:

   • The x-intercept is found by setting \(s = 0\) in the equation: \[ 1.50b + 0.50(0) = 20 \implies 1.50b = 20 \implies b = \frac{20}{1.50} \approx 13.33 \] So, the x-intercept is approximately \((13.33, 0)\).

   • The y-intercept is found by setting \(b = 0\) in the equation: \[ 1.50(0) + 0.50s = 20 \implies 0.50s = 20 \implies s = \frac{20}{0.50} = 40 \] So, the y-intercept is \((0, 40)\).

2. Plotting the Points:

   • You should plot the two intercepts:
     • Point A at \((13.33, 0)\) which is on the x-axis.
     • Point B at \((0, 40)\) which is on the y-axis.

3. Draw the Line:

   • Use a ruler to connect the two points (A and B). The line should extend in both directions through these two points.
   • Shade or draw arrows on both ends to indicate that the line continues.

Graph Description:

• The line represents all combinations of \(b\) (board books) and \(s\) (soft cover books) that Santiago can buy for exactly $20.
• The slope of the line is negative because as Santiago buys more board books (\(b\)), the number of soft cover books (\(s\)) he can buy decreases.

Note on the Range of \(b\) and \(s\):

When graphing, \(b\) and \(s\) must be non-negative since Santiago cannot buy a negative number of books. Therefore, for practical purposes, you should only consider the section of the line in the first quadrant (where both \(b \geq 0\) and \(s \geq 0\)).

Example Plotting:

1. \( b = 0 \) gives \( s = 40 \) (the point \((0, 40)\)).
2. \( b = 13.33 \) gives \( s = 0 \) (the point \((13.33, 0)\)).

Possible Visual:

Your graph will visually look something like this (drawn in the first quadrant):

s
|
40| * (0, 40)
  |
  |   
  |      
  |           
  |               
  |                  
  |                    *
  |                     (13.33, 0)
  |__________________________ b
   0        5       10     15

This line intersects the axes at points described and represents the solutions to the equation. You can then add any constraints given in your scenario, such as only purchasing whole books or potential maximums based on real-life context.

Answer 8

what would it look like on graph paper

Answer 9

To visualize what the graph of the equation \(1.50b + 0.50s = 20\) looks like on graph paper, you can follow these instructions. Below, I’ll describe how it would appear based on the coordinates we’ve calculated.

Step-by-Step Visualization on Graph Paper

1. Set Up the Axes:

   • X-axis (horizontal): Label it as "Number of Board Books (\(b\))".
   • Y-axis (vertical): Label it as "Number of Soft Cover Books (\(s\))".
   • Make sure to mark the origin (0, 0) at the bottom-left corner.

2. Scale:

   • Choose a scale for your axes. For this example, you could use 1 unit on the graph paper for every 1 board book or 1 soft cover book.
   • However, since the intercept points are around \( (13.33, 0) \) and \( (0, 40) \), you might want to use a slightly larger scale, such as:
     • 1 unit = 2 board books (this means point \( (13.33, 0) \) will be around 6.5 on the graph).
     • 1 unit = 5 soft cover books (this means point \( (0, 40) \) will be at 8 on the graph).

3. Plotting the Intercepts:

   • For the y-intercept: Plot the point \((0, 40)\) (adjusted down to 8 on your graph if you used the 5/1 scale).
   • For the x-intercept: Plot the point \((13.33, 0)\) (adjusted down to about 6.5 on your graph if using the 2/1 scale).

Graphing the Line:

• Draw the Line:
  • Using a ruler, draw a straight line between these two points.
  • Extend the line beyond the intercepts, and make sure to put arrows on each end of the line to indicate that it continues infinitely.

Example Representation on Graph Paper:

Here's a textual representation of how it would look:

   s
40 |        * (0, 40)
   |        
35 |        
   |        
30 |        
   |        
25 |        
   |        
20 |        
   |        
15 |        
   |        
10 |        
   |        
 5 |             
   |                  * (13.33,0)
 0 |_____________________________ b
   0       5        10       15

Key Features:

• Line: The downward sloping line connects the points and represents the combinations of board and soft cover books.
• Axes: Clearly labeled, and you can add tick marks for clarity.
• Arrowheads on the line: Indicating that the line extends into negative values (though they don't make sense in this context since you can't purchase negative books).

Final Thoughts:

Depending on the actual scale you choose and the grid size of your paper, the coordinates may vary slightly on your graph, but the essential features and relative positions of the x- and y-intercepts and the line connecting them will remain the same. This linear representation of the equation visually conveys the relationship between the number of board books and soft cover books purchased under the given financial constraint.

Answer 10

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.

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 11

To graph the equation \(1.50b + 0.50s = 20\) on graph paper and represent all the solutions without additional constraints, follow these simple steps:

Steps to Graph the Equation

1. Rewrite the Equation: As mentioned earlier, we can rearrange the original equation into a slope-intercept form: \[ s = 40 - 3b \] This means that for every board book (\(b\)) bought, the number of soft cover books (\(s\)) changes accordingly.

2. Identify the Intercepts:

   • X-Intercept (where \(s = 0\)): \[ 1.50b + 0.50(0) = 20 \implies 1.50b = 20 \implies b = \frac{20}{1.50} \approx 13.33 \]

   • Y-Intercept (where \(b = 0\)): \[ 1.50(0) + 0.50s = 20 \implies 0.50s = 20 \implies s = \frac{20}{0.50} = 40 \]

3. Plotting Points:

   • Plot the y-intercept: \((0, 40)\)
   • Plot the x-intercept: \((13.33, 0)\)

4. Draw the Axes:

   • Use graph paper with an appropriate scale. For example:
     • Each square could represent 1 unit.
     • Mark horizontal increments (b-axis) to at least 15 (up to 20 to give some buffer space for the line).
     • Mark vertical increments (s-axis) up to at least 40.

5. Draw the Line:

   • Plot the points:
     • Point A at \((0, 40)\).
     • Point B at \((13.33, 0)\).
   • Use a straightedge or ruler to connect the points and extend the line beyond them with arrows on either end.

Example on Graph Paper

Here's a conceptual representation of what it would look like on graph paper:

  s
  40 | * (0, 40)
     |
  35 |
     |
  30 |
     |
  25 |
     |
  20 |
     |
  15 |
     |
  10 |
     |
   5 |                      *
     |                       (13.33, 0)
   0 |_____________________________ b
     0       5        10       15

Conclusion

• The line drawn through the points represents the combination of board books and soft cover books Santiago can purchase for a total cost of $20.
• All points on the line are valid solutions to the equation, thus forming the solution set.

Next Steps

At this point, you could explore additional constraints based on the scenario (such as requiring whole numbers for the number of books or limits on the number of each type of book Santiago can buy), but for now, this graph captures all potential combinations of board and soft cover books based on the provided equation.