Question

the concession stand at a school basketball tournament sells hot dogs, hamburgers, and chicken sandwiches. during one game, the stand sold 16 hot dogs, 14 hamburgers, and 8 chicken sandwiches for a total of $89.00. during a second game, the stand sold 10 hot dogs, 13 hamburgers, and 5 chicken sandwiches for a total of $66.25. during a third game, the stand sold 4 hot dogs, 7 hamburgers, and 7 chicken sandwiches for a total of $52.25. determine the price of each product.
the price of a hot dog is $□. the price of a hamburger is $□. the price of a chicken sandwich is $□.
(type integers or decimals.)

Explanation:

Step1: Define variables for prices

Let $h$ = price of a hot dog, $b$ = price of a hamburger, $c$ = price of a chicken sandwich.

Step2: Set up system of equations

From game 1: $16h + 14b + 8c = 89.00$
From game 2: $10h + 13b + 5c = 66.25$
From game 3: $4h + 7b + 7c = 52.25$

Step3: Simplify first equation

Divide game 1 equation by 2:
$8h + 7b + 4c = 44.50$

Step4: Eliminate $h$ and $b$

Subtract simplified game 1 from game 3:
$(4h + 7b + 7c) - (8h + 7b + 4c) = 52.25 - 44.50$
$-4h + 3c = 7.75$ --- Equation (A)

Step5: Multiply game 2 equation by 4

$40h + 52b + 20c = 265.00$

Step6: Multiply game 1 equation by 5

$80h + 70b + 40c = 445.00$

Step7: Eliminate $c$ from game 2/1

Subtract 2×Step5 from Step6:
$(80h + 70b + 40c) - 2(40h + 52b + 20c) = 445.00 - 2×265.00$
$80h + 70b + 40c - 80h - 104b - 40c = 445 - 530$
$-34b = -85$

Step8: Solve for $b$

$b = \frac{-85}{-34} = 2.50$

Step9: Substitute $b$ into simplified game1

$8h + 7(2.50) + 4c = 44.50$
$8h + 17.50 + 4c = 44.50$
$8h + 4c = 27.00$
Divide by 4: $2h + c = 6.75$ --- Equation (B)

Step10: Solve Equation (A) and (B)

Equation (A): $-4h + 3c = 7.75$
Multiply Equation (B) by 2: $4h + 2c = 13.50$
Add the two equations:
$(-4h + 3c) + (4h + 2c) = 7.75 + 13.50$
$5c = 21.25$
$c = \frac{21.25}{5} = 4.25$

Step11: Solve for $h$ using Equation (B)

$2h + 4.25 = 6.75$
$2h = 6.75 - 4.25 = 2.50$
$h = \frac{2.50}{2} = 1.25$

Answer:

The price of a hot dog is $\boldsymbol{1.25}$. The price of a hamburger is $\boldsymbol{2.50}$. The price of a chicken sandwich is $\boldsymbol{4.25}$.