Questions: David bought 3 DVDs for 4 books for 40 at a yard sale. Anna bought 1 DVD and 6 books for 18. How much did each DVD and book cost? 3x + 4y = 40 1x + 6y = 18

David bought 3 DVDs for 4 books for 40 at a yard sale. Anna bought 1 DVD and 6 books for 18. How much did each DVD and book cost? 3x + 4y = 40 1x + 6y = 18
Transcript text: David bought 3 DVDs for 4 books for $\$ 40$ at a yard sale. Anna bought 1 DVD and 6 books for $\$ 18$. How much did each DVD and book cost? \[ \begin{array}{l} 3 x+4 y=40 \\ 1 x+6 y=18 \\ \end{array} \]
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Solution

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Solution Steps

To solve the problem of determining the cost of each DVD and book, we can set up a system of linear equations based on the information given. We have two equations: one for David's purchase and one for Anna's purchase. By solving this system of equations, we can find the individual costs of a DVD and a book.

Step 1: Set Up the Equations

We start by defining the variables:

  • Let \( x \) be the cost of one DVD.
  • Let \( y \) be the cost of one book.

From the problem statement, we can set up the following equations based on David's and Anna's purchases:

  1. \( 3x + 4y = 40 \) (David's purchase)
  2. \( x + 6y = 18 \) (Anna's purchase)
Step 2: Solve the System of Equations

We solve the system of equations to find the values of \( x \) and \( y \). The solution yields:

  • \( x = 12 \)
  • \( y = 1 \)
Step 3: Interpret the Results

The results indicate that the cost of one DVD is \( 12 \) dollars and the cost of one book is \( 1 \) dollar.

Final Answer

The cost of each DVD and book is given by: \[ \boxed{x = 12, \, y = 1} \]

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