Write a two-variable equation to represent this situation. Be sure to define the variables you use in your equation. please

Answer 1

Sure! To create a two-variable equation, I'll need a specific situation to work with. Could you please provide details about the scenario you're thinking of?

Answer 2

Algabra: 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.

Answer 3

To represent the situation with a two-variable equation, we first need to define our variables:

• Let \( b \) represent the number of board books Santiago buys.
• Let \( s \) represent the number of soft cover books Santiago buys.

Now, we can create an equation based on the total amount Santiago spends. We know that each board book costs $1.50 and each soft cover book costs $0.50. Santiago spends a total of $20.

The equation that represents this situation is:

\[ 1.50b + 0.50s = 20 \]

In summary:

• \( b \) = number of board books
• \( s \) = number of soft cover books
• The equation \( 1.50b + 0.50s = 20 \) represents the total amount spent on books.

Answer 4

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 5

To check if it's possible that Santiago bought 10 board books and 5 soft cover books, we can substitute \( b = 10 \) and \( s = 5 \) into the equation:

\[ 1.50b + 0.50s = 20 \]

Substituting the values:

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

Calculating:

\[ 1.50 \times 10 = 15 \] \[ 0.50 \times 5 = 2.5 \] Now, adding these amounts together:

\[ 15 + 2.5 = 17.5 \]

Since \( 17.5 \) is not equal to \( 20 \), this means that it's not possible for Santiago to have bought 10 board books and 5 soft cover books if he spent exactly $20.

Explanation: The total cost of 10 board books and 5 soft cover books would be $17.50, which is less than the $20 he spent at the book sale. Therefore, he must have bought a different combination of books to reach the total of $20.

Answer 6

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 7

While I can't physically create a graph for you here, I can guide you through the process of graphing the solution set for the equation \( 1.50b + 0.50s = 20 \). Here’s how you can do it step by step on graph paper:

Step 1: Rewrite the Equation

First, let's rewrite the equation in slope-intercept form ( \( s = mx + b \) ).

Starting with:

\[ 1.50b + 0.50s = 20 \]

Rearranging gives:

\[ 0.50s = 20 - 1.50b \] \[ s = -3b + 40 \]

Step 2: Identify Points

To graph this equation, we can find two or more points that satisfy this equation by choosing values for \( b \) (number of board books) and calculating corresponding values for \( s \) (number of soft cover books).

Finding Points:

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

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

3. When \( b = 20 \): \[ s = -3(20) + 40 = -20 \quad \Rightarrow (20, -20) \quad \text{(not a valid point since number of books can't be negative)} \]

This gives us the points \( (0, 40) \) and \( (10, 10) \).

Step 3: Plot the Points

1. On your graph paper, plot the points \( (0, 40) \) and \( (10, 10) \).

Step 4: Draw the Line

Using a ruler, draw a straight line through the points you plotted. This line represents the solution set of the equation \( 1.50b + 0.50s = 20 \).

Step 5: Interpret the Graph

The graph you created shows all the possible combinations of board books (b) and soft cover books (s) that Santiago could buy if there were no other constraints (like the number of books being non-negative).

Note:

Since \( b \) and \( s \) represent quantities (number of books), the solution set must also consider \( b \geq 0 \) and \( s \geq 0 \). Thus, only the portion of the line in the first quadrant (where both \( b \) and \( s \) are non-negative) would be relevant to the original scenario. You can highlight or indicate this portion on your graph.

Answer 8

make it copy and paste it?

Answer 9

Certainly! Below is a simple representation of the equation \(1.50b + 0.50s = 20\) using ASCII art for a graph.

s
|
|                                         *
|                                   *
|                             *
|                       *
|                 *
|           *
|       *
|   *
|________________________________________ b
0   2   4   6   8   10  12  14  16  18  20

Points Explained:

• The * symbol represents points plotted based on the equation.
• From the calculation we did earlier, we plotted the points:
  • (0, 40) is significantly above the graph area if you were working with an actual graph where \(b\) and \(s\) can go much higher.
  • (10, 10) is plotted as an example.

Simplified Points to Plot:

For the sake of simplicity, you might focus on plotting points such as:

• (0, 40)
• (10, 10)

Important Notes:

• The graph should ideally be on graph paper or using a graphing tool for accuracy.
• Remember to consider only the first quadrant (where both \(b\) and \(s\) are non-negative) while drawing the line that connects the plotted points.

Method to Create in Excel or Online Graphing Tools:

If you want to create a more accurate graph:

1. Use an online graphing calculator (like Desmos).
2. Enter the equation \(0.5s + 1.5b = 20\) or rearranged to \(s = -3b + 40\) to see the plot visually.
3. Highlight the relevant first-quadrant part for the number of books.

Feel free to ask if you would like more specific formatting or details!

Answer 10

What are the constraints of the situation? Describe them.

Answer 11

In the context of the situation involving Santiago purchasing books for the day care, the constraints can be defined based on practical and logical considerations. Here are the main constraints:

1. Non-negativity of Variables:

• Number of Board Books (b): \( b \geq 0 \)

  • Santiago cannot buy a negative number of board books. If he buys none, \( b \) would be zero.

• Number of Soft Cover Books (s): \( s \geq 0 \)

  • Similarly, Santiago cannot buy a negative number of soft cover books. Buying none would also mean \( s \) is zero.

2. Budget Constraint:

• Total Spending: \( 1.50b + 0.50s \leq 20 \)
  • Santiago cannot spend more than $20. This includes the total cost of the board books and soft cover books. He must stay within this budget.

3. Practical Considerations:

• Maximum Number of Books:

  • Although not explicitly given, there may be a practical maximum number of books he could realistically buy based on the sale capacity and availability of books.

• Usefulness of Books:

  • The books Santiago chooses should ideally be beneficial or appropriate for the day care to be useful. This adds a qualitative constraint on the selection of books.

Summary of Constraints:

1. \( b \geq 0 \): No negative board books.
2. \( s \geq 0 \): No negative soft cover books.
3. \( 1.50b + 0.50s \leq 20 \): Total spending must not exceed $20.
4. Practical and qualitative considerations regarding usefulness and availability of books.

When graphing the solution set, these constraints help define the feasible region where Santiago's choices lie, usually limited to the first quadrant of the graph where both variables are non-negative and below the line defined by the budget equation.

Answer 12

make it copy and paste able

Answer 13

Certainly! Below is a copy-and-paste friendly format of the constraints for the situation:

Constraints of the Situation:

1. Non-negativity of Variables:
   - Number of Board Books (b): b ≥ 0
   - Number of Soft Cover Books (s): s ≥ 0

2. Budget Constraint:
   - Total Spending: 1.50b + 0.50s ≤ 20

3. Practical Considerations:
   - Maximum Number of Books: While not explicitly given, there may be a practical maximum based on the sale capacity and availability of books.
   - Usefulness of Books: The books chosen should be beneficial or appropriate for the day care.

Summary of Constraints:
1. b ≥ 0: No negative board books.
2. s ≥ 0: No negative soft cover books.
3. 1.50b + 0.50s ≤ 20: Total spending must not exceed $20.
4. Considerations regarding usefulness and availability of books.

Feel free to copy and paste this as needed!

Answer 14

Use your graph to help you make a list of all viable solutions to the problem.
copy and paste able

Answer 15

To determine viable solutions based on the graph of the equation \(1.50b + 0.50s = 20\) and the constraints we established, we would typically look for integer points (since the number of books must be a whole number) that lie on or below the line formed by the equation within the first quadrant.

Viable Solutions (Integer Points Only):

Here’s a list of viable solutions considering the equation and constraints:

Viable Solutions (b, s):

1. (0, 40)
2. (1, 37.5) - not viable (s must be an integer)
3. (2, 35)
4. (3, 32.5) - not viable
5. (4, 30)
6. (5, 27.5) - not viable
7. (6, 25)
8. (7, 22.5) - not viable
9. (8, 20)
10. (9, 17.5) - not viable
11. (10, 15)
12. (11, 12.5) - not viable
13. (12, 10)
14. (13, 7.5) - not viable
15. (14, 5)
16. (15, 2.5) - not viable
17. (16, 0)

Summary of Viable Integer Solutions:

1. (0, 40)
2. (2, 35)
3. (4, 30)
4. (6, 25)
5. (8, 20)
6. (10, 15)
7. (12, 10)
8. (14, 5)
9. (16, 0)

Note:

When listing these solutions, we focus only on pairs \((b, s)\) where both \(b\) and \(s\) are non-negative integers, which means we discard any points where the number of soft cover books is not an integer or would be negative. The viable solutions indicate the different combinations of board books and soft cover books Santiago can purchase while staying within the budget of $20.

Feel free to copy and paste the above text!

Answer 16

Suppose Santiago bought 34 books. Can you figure out how many of each type of book he bought? Explain:

Answer 17

Certainly! To find out how many of each type of book Santiago bought, we need to solve a system of equations based on the information given:

1. We know total books purchased: \[ b + s = 34 \] where \(b\) is the number of board books and \(s\) is the number of soft cover books.

2. We also know the total amount spent: \[ 1.50b + 0.50s = 20 \]

Step 1: Rearranging the First Equation

From the first equation, we can express \(s\) in terms of \(b\): \[ s = 34 - b \]

Step 2: Substituting into the Second Equation

Now, we'll substitute this expression for \(s\) into the second equation: \[ 1.50b + 0.50(34 - b) = 20 \]

Step 3: Simplifying the Second Equation

Distributing the \(0.50\) gives: \[ 1.50b + 17 - 0.50b = 20 \]

Now, combine the \(b\) terms: \[ (1.50 - 0.50)b + 17 = 20 \] \[ 1.00b + 17 = 20 \]

Step 4: Solving for \(b\)

Now, subtract \(17\) from both sides: \[ 1.00b = 3 \] \[ b = 3 \]

Step 5: Finding \(s\)

Now that we have \(b\), we can find \(s\) using the equation \(s = 34 - b\): \[ s = 34 - 3 = 31 \]

Conclusion

Santiago bought:

• 3 board books (b = 3)
• 31 soft cover books (s = 31)

Verification

To ensure the solution is correct, we can check:

1. Total number of books: \[ 3 + 31 = 34 \quad \text{(Correct)} \]

2. Total cost: \[ 1.50(3) + 0.50(31) = 4.50 + 15.50 = 20 \quad \text{(Correct)} \]

Thus, the solution fits both conditions, confirming that Santiago bought 3 board books and 31 soft cover books.

Answer 18

make it so I can copy and paste able

Answer 19

Sure! Here’s the complete explanation in a copy-and-paste format:

To find out how many of each type of book Santiago bought, we need to solve a system of equations based on the information given:

1. Total books purchased:
   b + s = 34
   where b is the number of board books and s is the number of soft cover books.

2. Total amount spent:
   1.50b + 0.50s = 20

Step 1: Rearranging the First Equation
From the first equation, we express s in terms of b:
   s = 34 - b

Step 2: Substituting into the Second Equation
Substituting this expression for s into the second equation:
   1.50b + 0.50(34 - b) = 20

Step 3: Simplifying the Second Equation
Distributing the 0.50 gives:
   1.50b + 17 - 0.50b = 20

Combining the b terms:
   (1.50 - 0.50)b + 17 = 20
   1.00b + 17 = 20

Step 4: Solving for b
Subtracting 17 from both sides:
   1.00b = 3
   b = 3

Step 5: Finding s
Now that we have b, we can find s using the equation s = 34 - b:
   s = 34 - 3 = 31

Conclusion
Santiago bought:
- 3 board books (b = 3)
- 31 soft cover books (s = 31)

Verification
1. Total number of books:
   3 + 31 = 34 (Correct)

2. Total cost:
   1.50(3) + 0.50(31) = 4.50 + 15.50 = 20 (Correct)

Thus, the solution fits both conditions, confirming that Santiago bought 3 board books and 31 soft cover books.

Feel free to copy and use this text as needed!