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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 hard cover books are $2.00 and soft cover books are $1.00. He picks out a variety of books and spends $30. F. Suppose Santiago bought 26 books. Can you figure out how many of each type of book he bought? Explain. Write a system of equations and solve algebraically. Use Demos Calculator to check your work by adding the equation x + y = 26 to the graph above to help you answer the question

1 answer

To solve the problem, we can define two variables:

  • Let \( x \) represent the number of hardcover books Santiago bought.
  • Let \( y \) represent the number of softcover books Santiago bought.

From the problem, we know two things:

  1. Santiago bought a total of 26 books: \[ x + y = 26 \]

  2. Santiago spent a total of $30, with hardcover books costing $2.00 each and softcover books costing $1.00 each: \[ 2x + 1y = 30 \]

Now we can set up the system of equations: \[

  1. \quad x + y = 26 \quad \text{(Equation 1)} \] \[
  2. \quad 2x + y = 30 \quad \text{(Equation 2)} \]

Step 1: Solve the System of Equations

We can solve this system using the substitution or elimination method. In this case, we'll use the substitution method.

From Equation 1, we can express \( y \) in terms of \( x \): \[ y = 26 - x \]

Now we substitute \( y \) in Equation 2: \[ 2x + (26 - x) = 30 \]

Step 2: Simplify and Solve for \( x \)

\[ 2x + 26 - x = 30 \] \[ x + 26 = 30 \] \[ x = 30 - 26 \] \[ x = 4 \]

Step 3: Find \( y \)

Now that we have \( x \), we can find \( y \): \[ y = 26 - x = 26 - 4 = 22 \]

Conclusion

Santiago bought:

  • \( x = 4 \) hardcover books
  • \( y = 22 \) softcover books

Verification

Now let's verify if these numbers satisfy both original equations:

  1. Total number of books: \[ x + y = 4 + 22 = 26 \quad \text{(Correct)} \]

  2. Total cost: \[ 2x + 1y = 2(4) + 1(22) = 8 + 22 = 30 \quad \text{(Correct)} \]

Final Answer

Santiago bought 4 hardcover books and 22 softcover books.