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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 it, but simplify the answer

5 answers

To graph the situation, we can define two variables:

  • \( x \): the number of board books Santiago buys
  • \( y \): the number of soft cover books Santiago buys

The cost information gives us the equations:

  • The cost of the board books is \( 1.50x \)
  • The cost of the soft cover books is \( 0.50y \)

Given that Santiago spends a total of $20, we can set up the equation:

\[ 1.50x + 0.50y = 20 \]

To simplify this equation, we can multiply all terms by 2 to eliminate the decimal:

\[ 3x + y = 40 \]

Rearranging the Equation

Now we can rearrange this equation to express \( y \) in terms of \( x \):

\[ y = 40 - 3x \]

Finding Intercepts

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

  • x-intercept: Set \( y = 0 \) in the equation. \[ 40 - 3x = 0 \implies 3x = 40 \implies x = \frac{40}{3} \approx 13.33 \]

  • y-intercept: Set \( x = 0 \) in the equation. \[ y = 40 - 3(0) = 40 \]

Points for the Graph

Thus, we have two points:

  1. The x-intercept: \( (13.33, 0) \)
  2. The y-intercept: \( (0, 40) \)

Graphing the Line

You can plot these points on a Cartesian plane, and draw a line through them. The line represents all combinations of board books and soft cover books that Santiago could buy for $20.

Domain and Range

Since both \( x \) and \( y \) represent the number of books, they must be non-negative. Therefore, both \( x \) and \( y \) must be greater than or equal to 0.

  • \( 0 \leq x \leq \frac{40}{3} \)
  • \( 0 \leq y \leq 40 \)

Conclusion

The graph of the equation \( 3x + y = 40 \) will show the relationship between the number of board books and the number of soft cover books that Santiago can purchase within his budget of $20.

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

To find all viable solutions to the problem of how many board books (\(x\)) and soft cover books (\(y\)) Santiago can buy for $20, we'll look for non-negative integer solutions that satisfy the equation derived from the previous steps:

\[ 3x + y = 40 \]

Finding Integer Solutions

To generate integer solutions, we can start by substituting different integer values for \(x\) and calculating the corresponding \(y\) values while keeping both \(x\) and \(y\) as non-negative integers.

Step-by-step calculations:

  1. For \(x = 0\): \[ y = 40 - 3(0) = 40 \quad \text{(Solution: (0, 40))} \]

  2. For \(x = 1\): \[ y = 40 - 3(1) = 37 \quad \text{(Solution: (1, 37))} \]

  3. For \(x = 2\): \[ y = 40 - 3(2) = 34 \quad \text{(Solution: (2, 34))} \]

  4. For \(x = 3\): \[ y = 40 - 3(3) = 31 \quad \text{(Solution: (3, 31))} \]

  5. For \(x = 4\): \[ y = 40 - 3(4) = 28 \quad \text{(Solution: (4, 28))} \]

  6. For \(x = 5\): \[ y = 40 - 3(5) = 25 \quad \text{(Solution: (5, 25))} \]

  7. For \(x = 6\): \[ y = 40 - 3(6) = 22 \quad \text{(Solution: (6, 22))} \]

  8. For \(x = 7\): \[ y = 40 - 3(7) = 19 \quad \text{(Solution: (7, 19))} \]

  9. For \(x = 8\): \[ y = 40 - 3(8) = 16 \quad \text{(Solution: (8, 16))} \]

  10. For \(x = 9\): \[ y = 40 - 3(9) = 13 \quad \text{(Solution: (9, 13))} \]

  11. For \(x = 10\): \[ y = 40 - 3(10) = 10 \quad \text{(Solution: (10, 10))} \]

  12. For \(x = 11\): \[ y = 40 - 3(11) = 7 \quad \text{(Solution: (11, 7))} \]

  13. For \(x = 12\): \[ y = 40 - 3(12) = 4 \quad \text{(Solution: (12, 4))} \]

  14. For \(x = 13\): \[ y = 40 - 3(13) = 1 \quad \text{(Solution: (13, 1))} \]

  15. For \(x = 14\) (not valid, as \(y\) becomes negative): \[ y = 40 - 3(14) = -2 \quad \text{(not valid)} \]

List of Viable Solutions

Thus, the viable combinations of board books (\(x\)) and soft cover books (\(y\)) that Santiago can buy for $20 are:

  1. (0, 40)
  2. (1, 37)
  3. (2, 34)
  4. (3, 31)
  5. (4, 28)
  6. (5, 25)
  7. (6, 22)
  8. (7, 19)
  9. (8, 16)
  10. (9, 13)
  11. (10, 10)
  12. (11, 7)
  13. (12, 4)
  14. (13, 1)

These pairs represent all the different combinations of board and soft cover books Santiago can purchase without exceeding the budget of $20.

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

If Santiago bought a total of 34 books, we can use the equations we derived earlier to find out how many board books (\(x\)) and soft cover books (\(y\)) he bought.

We have two equations:

  1. The total cost equation: \[ 1.50x + 0.50y = 20 \] This can be simplified to: \[ 3x + y = 40 \quad \text{(after multiplying by 2)} \]

  2. The total number of books equation: \[ x + y = 34 \]

Solving the Two Equations

We now have a system of equations:

  1. \( 3x + y = 40 \)
  2. \( x + y = 34 \)

Step 1: Isolate \(y\) in the second equation: \[ y = 34 - x \]

Step 2: Substitute \(y\) in the first equation: Now we substitute \(y\) from the second equation into the first equation: \[ 3x + (34 - x) = 40 \]

Step 3: Simplify the equation: \[ 3x + 34 - x = 40 \] \[ 2x + 34 = 40 \] \[ 2x = 40 - 34 \] \[ 2x = 6 \] \[ x = 3 \]

Step 4: Find \(y\): Now that we have \(x\), we can find \(y\) using the equation for \(y\): \[ y = 34 - x \] \[ y = 34 - 3 = 31 \]

Conclusion

Santiago bought:

  • \(3\) board books
  • \(31\) soft cover books

This solution satisfies both the total number of books and the total cost, as we can verify:

  • Total number of books: \(3 + 31 = 34\) (correct)
  • Total cost: \(1.50(3) + 0.50(31) = 4.50 + 15.50 = 20\) dollars (correct)

Thus, he purchased 3 board books and 31 soft cover books.