Question

write a system of equations to model each situation. solve the system by substitution. check the solution. 15. todd purchased two cds and three cassette tapes for $75. paulena went to the same music store and purchased one cd and five cassette tapes for $76. if all cds are the same price and all the cassette tapes are the same price, find the cost of each cd and each cassette tape.

Explanation:

Step1: Define Variables

Let \( x \) be the cost of a CD (in dollars) and \( y \) be the cost of a cassette tape (in dollars).

Step2: Set Up Equations

From Todd's purchase: \( 2x + 3y = 75 \)
From Paulena's purchase: \( x + 5y = 76 \)

Step3: Solve for \( x \) from Second Equation

From \( x + 5y = 76 \), we get \( x = 76 - 5y \).

Step4: Substitute \( x \) into First Equation

Substitute \( x = 76 - 5y \) into \( 2x + 3y = 75 \):
\( 2(76 - 5y) + 3y = 75 \)
Expand: \( 152 - 10y + 3y = 75 \)
Simplify: \( 152 - 7y = 75 \)
Subtract 152: \( -7y = 75 - 152 = -77 \)
Divide by -7: \( y = \frac{-77}{-7} = 11 \)

Step5: Find \( x \) Using \( y = 11 \)

Substitute \( y = 11 \) into \( x = 76 - 5y \):
\( x = 76 - 5(11) = 76 - 55 = 21 \)

Step6: Check the Solution

For Todd: \( 2(21) + 3(11) = 42 + 33 = 75 \) (matches)
For Paulena: \( 21 + 5(11) = 21 + 55 = 76 \) (matches)

Answer:

The cost of each CD is \$21 and the cost of each cassette tape is \$11.