Question

1. at a sale on winter clothing, cody bought two pairs of gloves and four hats for $43.00. tori bought two pairs of gloves and two hats for $30.00. find the prices of the hats and gloves.

variable x
variable y
equation 1
equation 2

1. on monday joe bought 10 cups of coffee and 5 doughnuts for his office at the cost of $16.50. it turns out that the doughnuts were more popular than the coffee. on tuesday he bought 5 cups of coffee and 10 doughnuts for a total of $14.25. how much was each cup of coffee?

variable x
variable y
equation 1
equation 2

1. tickets to a movie cost $7.25 for adults and $5.50 for students. a group of friends purchased 8 tickets for $52.75. determine the total number of adult and student tickets.

variable x
variable y
equation 1
equation 2

Explanation:

Problem 3

Step1: Define Variables

Let \( x \) be the price of a pair of gloves (in dollars) and \( y \) be the price of a hat (in dollars).

Step2: Form Equations

From Cody's purchase: \( 2x + 4y = 43 \) (Equation 1)
From Tori's purchase: \( 2x + 2y = 30 \) (Equation 2)

Step3: Solve the System

Subtract Equation 2 from Equation 1:
\( (2x + 4y) - (2x + 2y) = 43 - 30 \)
\( 2y = 13 \)
\( y = 6.5 \)
Substitute \( y = 6.5 \) into Equation 2:
\( 2x + 2(6.5) = 30 \)
\( 2x + 13 = 30 \)
\( 2x = 17 \)
\( x = 8.5 \)

Step1: Define Variables

Let \( x \) be the price of a cup of coffee (in dollars) and \( y \) be the price of a doughnut (in dollars).

Step2: Form Equations

From Monday's purchase: \( 10x + 5y = 16.50 \) (Equation 1)
From Tuesday's purchase: \( 5x + 10y = 14.25 \) (Equation 2)

Step3: Solve the System

Multiply Equation 2 by 2: \( 10x + 20y = 28.50 \) (Equation 3)
Subtract Equation 1 from Equation 3:
\( (10x + 20y) - (10x + 5y) = 28.50 - 16.50 \)
\( 15y = 12 \)
\( y = 0.8 \)
Substitute \( y = 0.8 \) into Equation 1:
\( 10x + 5(0.8) = 16.50 \)
\( 10x + 4 = 16.50 \)
\( 10x = 12.50 \)
\( x = 1.25 \)

Step1: Define Variables

Let \( x \) be the number of adult tickets and \( y \) be the number of student tickets.

Step2: Form Equations

Total tickets: \( x + y = 8 \) (Equation 1)
Total cost: \( 7.25x + 5.50y = 52.75 \) (Equation 2)

Step3: Solve the System

From Equation 1: \( y = 8 - x \)
Substitute \( y = 8 - x \) into Equation 2:
\( 7.25x + 5.50(8 - x) = 52.75 \)
\( 7.25x + 44 - 5.50x = 52.75 \)
\( 1.75x = 8.75 \)
\( x = 5 \)
Then \( y = 8 - 5 = 3 \)

Answer:

• Variable \( x \): Price of a pair of gloves (\$8.50)
• Variable \( y \): Price of a hat (\$6.50)
• Equation 1: \( 2x + 4y = 43 \)
• Equation 2: \( 2x + 2y = 30 \)
• Prices: Gloves cost \$8.50 per pair, hats cost \$6.50 each.

Problem 4