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A. Writing the Two-Variable Equation
Let's define the variables:

Let
= the number of hard cover books Santiago buys.
Let
= the number of soft cover books Santiago buys.
The total cost of the books can be represented by the equation:
or, simplified,

B. Checking the Values for 10 Hard Cover and 5 Soft Cover Books
To check if it's possible for Santiago to buy 10 hard cover books and 5 soft cover books, we substitute
and
into the equation:
Calculating this gives:

Since
, this means it is NOT possible for Santiago to buy 10 hard cover books and 5 soft cover books because it does not satisfy the equation.

C. Graphing the Solution Set
To graph the solution set for the equation
:

Find the intercepts:

x-intercept: Set
:
y-intercept: Set
:
Plot the points (0, 30) and (15, 0) on graph paper, and draw a straight line through these points. This line represents all combinations of hard cover and soft cover books that Santiago could buy for $30.

D. Constraints of the Situation
The constraints of this situation can be described as follows:

Non-negativity: Santiago cannot buy a negative number of books. Thus:

(cannot buy negative hard cover books)
(cannot buy negative soft cover books)
Whole Number Books: Santiago can only purchase whole books, meaning that
and
must be whole numbers (integers).

Limited Budget: The total cost must not exceed $30, which is already represented by the equation
.

Putting these constraints together, we can summarize them:

and
are integers.
These constraints limit the feasible solutions to the original equation, ensuring that they reflect the real-world context of the problem.
E. Use your graph to help you list three viable solutions to the problem when 2.00x + 1. 00y = 30.

1 answer

To find viable solutions to the equation \(2.00x + 1.00y = 30\) where \(x\) is the number of hard cover books and \(y\) is the number of soft cover books, we can utilize the information from the graph and also perform some calculations to find combinations of \(x\) and \(y\) that satisfy the equation while adhering to the constraints identified earlier.

Finding Viable Solutions

We can express \(y\) in terms of \(x\) from the equation:

\[ y = 30 - 2x \]

Now, substituting different integer values for \(x\) will help us find corresponding integer values for \(y\).

  1. Solution 1:

    • Let \(x = 0\): \[ y = 30 - 2(0) = 30 \] Viable Solution: \((0, 30)\) [0 hard cover books, 30 soft cover books]

  2. Solution 2:

    • Let \(x = 10\): \[ y = 30 - 2(10) = 10 \] Viable Solution: \((10, 10)\) [10 hard cover books, 10 soft cover books]

  3. Solution 3:

    • Let \(x = 15\): \[ y = 30 - 2(15) = 0 \] Viable Solution: \((15, 0)\) [15 hard cover books, 0 soft cover books]

Summary of Viable Solutions

Based on our calculations, three viable solutions to the given equation \(2.00x + 1.00y = 30\) are:

  1. \((0, 30)\): 0 hard cover and 30 soft cover books
  2. \((10, 10)\): 10 hard cover and 10 soft cover books
  3. \((15, 0)\): 15 hard cover and 0 soft cover books

These solutions adhere to the constraints of only purchasing whole numbers of books and staying within a budget of $30.