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 $107.00. during a second game, the stand sold 10 hot dogs, 13 hamburgers, and 5 chicken sandwiches for a total of $81.75. during a third game, the stand sold 4 hot dogs, 7 hamburgers, and 7 chicken sandwiches for a total of $60.25. 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

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

Step2: Set up system of equations

From the three games:

1. $16x + 14y + 8z = 107.00$
2. $10x + 13y + 5z = 81.75$
3. $4x + 7y + 7z = 60.25$

Step3: Simplify first equation

Divide equation 1 by 2:
$8x + 7y + 4z = 53.50$

Step4: Eliminate $x$ with equations 2 and 3

Multiply equation 3 by 2.5: $10x + 17.5y + 17.5z = 150.625$
Subtract equation 2:
$(10x-10x)+(17.5y-13y)+(17.5z-5z)=150.625-81.75$
$4.5y + 12.5z = 68.875$
Multiply by 2: $9y + 25z = 137.75$ (Equation 4)

Step5: Eliminate $x$ with equations 1 and 3

Multiply equation 3 by 4: $16x + 28y + 28z = 241.00$
Subtract equation 1:
$(16x-16x)+(28y-14y)+(28z-8z)=241.00-107.00$
$14y + 20z = 134.00$
Divide by 2: $7y + 10z = 67.00$ (Equation 5)

Step6: Solve for $y$ and $z$

Multiply Equation 5 by 2.5: $17.5y + 25z = 167.50$
Subtract Equation 4:
$(17.5y-9y)+(25z-25z)=167.50-137.75$
$8.5y = 29.75$
$y = \frac{29.75}{8.5} = 3.50$
Substitute $y=3.50$ into Equation 5:
$7(3.50) + 10z = 67.00$
$24.50 + 10z = 67.00$
$10z = 67.00 - 24.50 = 42.50$
$z = 4.25$

Step7: Solve for $x$

Substitute $y=3.50, z=4.25$ into Equation 3:
$4x + 7(3.50) + 7(4.25) = 60.25$
$4x + 24.50 + 29.75 = 60.25$
$4x + 54.25 = 60.25$
$4x = 60.25 - 54.25 = 6.00$
$x = 1.50$

Answer:

The price of a hot dog is $\$1.50$. The price of a hamburger is $\$3.50$. The price of a chicken sandwich is $\$4.25$.