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 $93.00. during a second game, the stand sold 10 hot dogs, 13 hamburgers, and 5 chicken sandwiches for a total of $68.75. during a third game, the stand sold 4 hot dogs, 7 hamburgers, and 7 chicken sandwiches for a total of $55.75. 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

Let $x$ = price of a hot dog, $y$ = price of a hamburger, $z$ = price of a chicken sandwich.

Step2: Set up system of equations

From game 1: $16x + 14y + 8z = 93.00$
From game 2: $10x + 13y + 5z = 68.75$
From game 3: $4x + 7y + 7z = 55.75$

Step3: Simplify first equation

Divide game 1 equation by 2:
$8x + 7y + 4z = 46.50$

Step4: Eliminate $x$ with game 2

Multiply simplified game 1 by 5, game 2 by 4:
$40x + 35y + 20z = 232.50$
$40x + 52y + 20z = 275.00$
Subtract: $17y = 42.50$

Step5: Solve for $y$

$y = \frac{42.50}{17} = 2.50$

Step6: Substitute $y$ into game 3

$4x + 7(2.50) + 7z = 55.75$
$4x + 17.50 + 7z = 55.75$
$4x + 7z = 38.25$

Step7: Substitute $y$ into simplified game1

$8x + 7(2.50) + 4z = 46.50$
$8x + 17.50 + 4z = 46.50$
$8x + 4z = 29.00$ → $2x + z = 7.25$ → $z = 7.25 - 2x$

Step8: Substitute $z$ into Step6 equation

$4x + 7(7.25 - 2x) = 38.25$
$4x + 50.75 - 14x = 38.25$
$-10x = -12.50$

Step9: Solve for $x$

$x = \frac{-12.50}{-10} = 1.25$

Step10: Solve for $z$

$z = 7.25 - 2(1.25) = 7.25 - 2.50 = 4.75$

Answer:

The price of a hot dog is $\$1.25$. The price of a hamburger is $\$2.50$. The price of a chicken sandwich is $\$4.75$.