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 $\$ 119.00$. During a second game, the stand sold 10 hot dogs, 13 hamburgers, and 5 chicken sandwiches for a total of $\$ 89.25$. During a third game, the stand sold 4 hot dogs, 7 hamburgers, and 7 chicken sandwiches for a total of $\$ 63.25$. Determine the price of each product.

The price of a hot dog is $\$ \square$. The price of a hamburger is $\$ \square$. The price of a chicken sandwich is $\$ \square$.
(Type integers or decimals.)

Step 1 ：Let's denote the price of a hot dog as H, the price of a hamburger as B, and the price of a chicken sandwich as C. From the problem, we have the following system of equations: \[16H + 14B + 8C = 119.00\] \[10H + 13B + 5C = 89.25\] \[4H + 7B + 7C = 63.25\]

Step 2 ：First, let's multiply equation (2) by 4 and equation (3) by 10, then subtract equation (3) from equation (2). This will eliminate H: \[40H + 52B + 20C = 357.00\] \[40H + 70B + 70C = 632.50\] Subtracting these gives: \[0H + 18B - 50C = -275.50\]

Step 3 ：Next, let's multiply equation (1) by 10 and equation (3) by 40, then subtract equation (1) from equation (3). This will also eliminate H: \[400H + 280B + 160C = 1190.00\] \[160H + 280B + 280C = 2530.00\] Subtracting these gives: \[-240H + 0B + 120C = 1340.00\] Dividing by -240 gives: \[H - 0.5C = -5.58\]

Step 4 ：Now we have a system of two equations with two variables B and C. We can solve this system by substitution or elimination. Let's use substitution: From equation (5), we can express C in terms of H: \[C = 2H + 11.16\]

Step 5 ：Substitute equation (6) into equation (4): \[18B - 50(2H + 11.16) = -275.50\] \[18B - 100H - 558 = -275.50\] \[18B - 100H = 282.50\] Divide by 18: \[B = 5.56H + 15.69\]

Step 6 ：Now we have the prices of the hot dog (H), hamburger (B), and chicken sandwich (C) in terms of H. Substitute equations (6) and (7) into any of the original equations, for example, equation (1): \[16H + 14(5.56H + 15.69) + 8(2H + 11.16) = 119.00\]

Step 7 ：Solving this equation gives: \[H = 1.25\]

Step 8 ：Substitute \(H = 1.25\) into equations (6) and (7) to find B and C: \[B = 5.56*1.25 + 15.69 = 22.45\] \[C = 2*1.25 + 11.16 = 13.66\]

Step 9 ：So, the price of a hot dog is \(\boxed{1.25}\), the price of a hamburger is \(\boxed{22.45}\), and the price of a chicken sandwich is \(\boxed{13.66}\).