Explanation:
To solve this problem, we can set up a system of equations based on the information given in the three scenarios. Let's denote the prices of hot dogs, hamburgers, and chicken sandwiches as follows:
Let:
H = Price of a hot dog
B = Price of a hamburger
C = Price of a chicken sandwich
We have the following information from the three scenarios:
1. 16H + 14B + 8C = 193
2. 10H + 13B + 5C = 68.75
3. 4H + 7B + 7C = 53.25
Now we can solve this system of equations to find the values of H, B, and C.
We can use a tool like substitution or elimination to solve this system. Let's use the elimination method. Let's start by multiplying the second equation by 2 so that we can eliminate the term with B:
2(10H + 13B + 5C) = 2(68.75)
20H + 26B + 10C = 137.5
Now, subtract the first equation from this modified second equation to eliminate B:
(20H + 26B + 10C) - (16H + 14B + 8C) = 137.5 - 193
4H + 12B + 2C = -55.5
Now we have a new equation:
4H + 12B + 2C = -55.5
Combine this equation with the third equation to eliminate B:
(4H + 12B + 2C) + (4H + 7B + 7C) = -55.5 + 53.25
8H + 19B + 9C = -2.25
Now we have a new equation:
8H + 19B + 9C = -2.25
Next, we can substitute this equation into the second equation to solve for C:
8H + 19B + 9C = -2.25
9C = -2.25 - 8H - 19B
C = (-2.25 - 8H - 19B) / 9
Now that we have the value of C, we can substitute it back into the third equation to solve for B. Once we have B, we can substitute its value into any of the equations to solve for H.
Keep in mind that due to rounding in the original values, the calculated prices might not be exact integers or decimals. Make sure to perform accurate calculations to get the closest possible values for H, B, and C.