Ask a New Question
Menu

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 hard cover books are $2.00 and soft cover books are $1.00. He picks out a variety of books and spends $30.
A. Write a two-variable equation to represent this situation. Be sure to define the variables you use in your equation.

11 answers

Let's define the variables first:

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

In this scenario, the costs of the books are as follows:

  • Each hard cover book costs $2.00.
  • Each soft cover book costs $1.00.

Since Santiago spends a total of $30, we can write the equation representing this situation as:

\[ 2x + 1y = 30 \]

This equation indicates that the total cost of the hard cover and soft cover books is $30.

B. Is it possible that Santiago bought 10 hard cover books and 5 soft cover books? Use your equation from part A to check. Explain your answer

To determine if Santiago could have bought 10 hard cover books and 5 soft cover books, we will substitute \( x = 10 \) (for hard cover books) and \( y = 5 \) (for soft cover books) into the equation we derived in part A:

The equation is: \[ 2x + y = 30 \]

Substituting the values: \[ 2(10) + 1(5) = 30 \]

Calculating: \[ 20 + 5 = 30 \]

Since the left side of the equation equals the right side (30 = 30), it is possible that Santiago could have bought 10 hard cover books and 5 soft cover books. Thus, the answer is yes, it is possible.

Graph the solution set for the first equation please

To graph the solution set of the equation \(2x + y = 30\), we can follow these steps:

  1. Rearrange the Equation: We can rewrite the equation in slope-intercept form \(y = mx + b\).

    Starting with: \[ 2x + y = 30 \] Subtract \(2x\) from both sides: \[ y = -2x + 30 \]

  2. Identify the Y-Intercept and Slope:

    • The y-intercept (\(b\)) is 30 (this means the line crosses the y-axis at the point (0, 30)).
    • The slope (\(m\)) is -2 (for every 1 unit increase in \(x\), \(y\) decreases by 2).

  3. Plot Points:

    • Start with the y-intercept: (0, 30).
    • To find another point, let’s find where the line crosses the x-axis (where \(y = 0\)): \[ 0 = -2x + 30 \implies 2x = 30 \implies x = 15 \] So, another point is (15, 0).

  4. Draw the Line:

    • Plot the points (0, 30) and (15, 0).
    • Draw a straight line through these points, extending in both directions.

The graph represents all combinations of hard cover and soft cover books that total $30. To visualize, you can sketch it on a graph with the x-axis representing hard cover books (\(x\)) and the y-axis representing soft cover books (\(y\)).

Graph Summary:

  • Line equation: \(y = -2x + 30\)
  • Points: (0, 30) and (15, 0) are on the line.
  • The shaded region would generally extend downwards from the line, covering non-negative values of \(x\) and \(y\) (considering that Santiago cannot buy a negative number of books).

You can also create a visual graph using graphing software or graphing tools for a more precise representation.

Can you make it shorter and less confusing, like dumb it down a little please

Sure! Here’s a simpler breakdown of how to graph the equation \(2x + y = 30\):

  1. Rearrange the Equation:

    • Start with the equation: \[ 2x + y = 30 \]
    • Isolate \(y\): \[ y = -2x + 30 \]

  2. Find Points to Plot:

    • Y-Intercept: Set \(x = 0\): \[ y = -2(0) + 30 = 30 \quad \text{(Point: (0, 30))} \]
    • X-Intercept: Set \(y = 0\): \[ 0 = -2x + 30 \implies 2x = 30 \implies x = 15 \quad \text{(Point: (15, 0))} \]

  3. Plot the Points:

    • Plot the points (0, 30) and (15, 0) on a graph.

  4. Draw the Line:

    • Connect these two points with a straight line. This line represents all possible combinations of hard cover and soft cover books that Santiago could buy for $30.

Summary:

  • Points to plot: (0, 30) and (15, 0)
  • Draw a line through these points to show solutions for \(2x + y = 30\).

Now you have the graph of the equation!

What are the constraints of this solution? Describe them

In the context of the problem regarding Santiago buying books, the solution \(2x + y = 30\) has some important constraints based on the real-world scenario. These constraints are:

  1. Non-Negative Number of Books:

    • Santiago cannot buy a negative number of books. Therefore, both \(x\) (number of hard cover books) and \(y\) (number of soft cover books) must be zero or greater: \[ x \geq 0 \quad \text{and} \quad y \geq 0 \]

  2. Budget Constraint:

    • The equation \(2x + y = 30\) represents the total amount spent on books. Santiago's total spending cannot exceed $30. So, the equation itself serves as a boundary in this sense.

  3. Integer Values:

    • Since books are bought in whole numbers, \(x\) and \(y\) must be integers (whole numbers). This means: \[ x \in \mathbb{Z}^+ \quad \text{and} \quad y \in \mathbb{Z}^+ \]
    • For example, Santiago cannot buy 2.5 hard cover books or 3.7 soft cover books.

Summary of Constraints:

  • \(x \geq 0\) (cannot buy negative hard cover books)
  • \(y \geq 0\) (cannot buy negative soft cover books)
  • \(x\) and \(y\) must be whole numbers (integers)

These constraints limit the possible combinations of hard cover and soft cover books that Santiago can purchase while adhering to the budget and practical aspects of buying books.

Can you make it way shorter please and just sum it all up please and thank you

Sure! Here’s a concise summary of the constraints:

  1. Non-Negative: Both \(x\) (hard cover books) and \(y\) (soft cover books) must be \( \geq 0\).
  2. Whole Numbers: \(x\) and \(y\) must be integers (you can't buy a fraction of a book).
  3. Budget Limit: The total spent must equal $30, as represented by the equation \(2x + y = 30\).

These constraints ensure Santiago buys a valid number of books within his budget.