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Question 1

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.

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

11 answers

To represent Santiago's book purchase situation, we can define the following variables:

  • Let \( x \) be the number of board books he buys.
  • Let \( y \) be the number of soft cover books he buys.

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.

We can write the equation representing this situation as:

\[ 1.50x + 0.50y = 20 \]

This equation expresses that the total cost of the board books (the term \( 1.50x \)) plus the total cost of the soft cover books (the term \( 0.50y \)) equals $20.

Is it possible that Santiago bought 10 board books and five soft cover books? Use your
equation from part A, 1.50x + 0.50y = 20, to check. Explain your answer.

To determine whether Santiago could have bought 10 board books and 5 soft cover books, we can substitute \( x = 10 \) and \( y = 5 \) into the equation \( 1.50x + 0.50y = 20 \) and see if the left-hand side equals 20.

Substituting the values:

\[ 1.50(10) + 0.50(5) = 15 + 2.50 = 17.50 \]

The left-hand side calculates to \( 17.50 \), which is not equal to \( 20 \).

Therefore, it is not possible that Santiago bought 10 board books and 5 soft cover books, since the total cost would only be $17.50, which is less than the $20 he spent.

Graph the equation, 1.50x + 0.50y = 20.

To graph the equation \(1.50x + 0.50y = 20\), we can follow these steps:

Step 1: Rewrite the Equation in Slope-Intercept Form (if necessary)

First, we can solve for \(y\) in terms of \(x\):

\[ 1.50x + 0.50y = 20 \]

Subtract \(1.50x\) from both sides:

\[ 0.50y = 20 - 1.50x \]

Now divide everything by \(0.50\):

\[ y = 40 - 3x \]

Step 2: Identify Intercepts

To graph the equation, we can find the x-intercept and the y-intercept.

  1. Finding the x-intercept (where \(y = 0\)): \[ 0 = 40 - 3x \implies 3x = 40 \implies x = \frac{40}{3} \approx 13.33 \]

  2. Finding the y-intercept (where \(x = 0\)): \[ y = 40 - 3(0) = 40 \]

Step 3: Plot the Points

  • The x-intercept is approximately \((13.33, 0)\).
  • The y-intercept is \((0, 40)\).

Step 4: Draw the Graph

  • Draw a coordinate system with \(x\) and \(y\) axes.
  • Plot the points \((13.33, 0)\) and \((0, 40)\).
  • Draw a straight line through these two points to represent the equation \(1.50x + 0.50y = 20\).

Step 5: Indicate the Relevant Quadrant

Since the context of the problem involves buying books, only the first quadrant (where \(x \geq 0\) and \(y \geq 0\)) is relevant.

Final Notes

The graph will be a downward-sloping line indicating the relationship between the number of board books (\(x\)) and the number of soft cover books (\(y\)) that Santiago can buy with a budget of $20.

Keep in mind that you might want to represent points \( (x, y) \) as whole numbers, since you can’t buy a fraction of a book. Thus, you will be looking at feasible integer solutions in the graph.

So what is the answer?

To summarize the answer for graphing the equation \(1.50x + 0.50y = 20\):

  1. Convert to Slope-Intercept Form: The equation can be rewritten as: \[ y = 40 - 3x \]

  2. Find the Intercepts:

    • X-intercept:
      • Set \(y = 0\): \[ 0 = 40 - 3x \implies x = \frac{40}{3} \approx 13.33 \] So the x-intercept is approximately \((13.33, 0)\).
    • Y-intercept:
      • Set \(x = 0\): \[ y = 40 - 3(0) = 40 \] So the y-intercept is \((0, 40)\).

  3. Plot the Points:

    • Plot the intercepts on a graph:
      • X-intercept: \((13.33, 0)\)
      • Y-intercept: \((0, 40)\)

  4. Draw the Line:

    • Draw a straight line through the two points, creating a downward-sloping line from the y-intercept to the x-intercept.

  5. Consider Feasible Solutions:

    • Since \(x\) (the number of board books) and \(y\) (the number of soft cover books) must be whole numbers, you would look for integer solutions along this line in the first quadrant where both values are non-negative.

Conclusion:

The graph of the equation \(1.50x + 0.50y = 20\) will intersect the x-axis at approximately \((13.33, 0)\) and the y-axis at \((0, 40)\). Only points with non-negative integer \(x\) and \(y\) values along this line are valid in the context of Santiago’s book purchases.

If you need a visual representation, you would need to use graphing software or graph paper, as I am unable to create images directly.

What are the constraints of this situation? Describe them

In this situation involving Santiago buying books, there are several important constraints to consider. These constraints will help define the feasible region for the number of board books and soft cover books he can purchase. Here are the key constraints:

  1. Non-negativity Constraints:

    • Since Santiago cannot buy a negative number of books, the number of board books (\(x\)) and the number of soft cover books (\(y\)) must both be greater than or equal to zero: \[ x \geq 0 \] \[ y \geq 0 \]

  2. Budget Constraint:

    • Santiago has a budget limitation. The total amount spent on both types of books cannot exceed $20. This is already represented by the original equation: \[ 1.50x + 0.50y \leq 20 \]
    • In practical terms, this means that the cost of board books at $1.50 each plus the cost of soft cover books at $0.50 each cannot exceed the total of $20.

  3. Integer Constraints:

    • In the context of purchasing books, \(x\) and \(y\) must be non-negative integers. This implies: \[ x \in \mathbb{Z}^+ \quad (x = 0, 1, 2, \ldots) \] \[ y \in \mathbb{Z}^+ \quad (y = 0, 1, 2, \ldots) \]
    • This means that while \(x\) and \(y\) can be any whole number, they cannot take on fractional values, as he cannot buy a fraction of a book.

Summary of Constraints:

  • \(x \geq 0\): Number of board books cannot be negative.
  • \(y \geq 0\): Number of soft cover books cannot be negative.
  • \(1.50x + 0.50y \leq 20\): Total spending cannot exceed $20.
  • \(x \in \mathbb{Z}^+\): Number of board books must be a non-negative integer.
  • \(y \in \mathbb{Z}^+\): Number of soft cover books must be a non-negative integer.

These constraints create a bounded feasible region on a graph, where realistic combinations of board books and soft cover books can be observed within the limits of Santiago's budget.

Simplify the answer

Certainly! Here’s a simplified summary of the constraints for Santiago's book purchasing situation:

Constraints:

  1. Non-negativity:

    • Board Books: \(x \geq 0\) (number of board books cannot be negative)
    • Soft Cover Books: \(y \geq 0\) (number of soft cover books cannot be negative)

  2. Budget Constraint:

    • The total cost cannot exceed $20: \[ 1.50x + 0.50y \leq 20 \]

  3. Integer Values:

    • Both \(x\) (number of board books) and \(y\) (number of soft cover books) must be non-negative integers: \[ x \in \mathbb{Z}^+, \quad y \in \mathbb{Z}^+ \]

Summary:

  • \(x \geq 0\)
  • \(y \geq 0\)
  • \(1.50x + 0.50y \leq 20\)
  • \(x, y \in \mathbb{Z}^+\) (non-negative integers)

These constraints define the realistic possibilities for the quantities of books Santiago can purchase.