Final answer:
The rate of change is the cost of one hot dog, $2.50, and the initial value is the cost of a soda, $2. The student is dealing with a system of linear equations to determine these values.
Step-by-step explanation:
The student is asking about a system of equations related to the cost of sodas and hot dogs. The two purchases made, one for a soda and 2 hot dogs at $7, and another for a soda and 4 hot dogs at $12, help us define a system of linear equations to solve for the individual prices of a soda and a hot dog.
Let's denote the price of a soda as 's' and the price of a hot dog as 'h'. The first customer's purchase gives us the equation:
s + 2h = 7
The second customer's purchase gives us the equation:
s + 4h = 12
Subtracting the first equation from the second, we get:
2h = 5
From this, we find that h = 2.5.
Rate of change here refers to the change in cost per additional hot dog, which is the price of one hot dog, $2.50. The initial value would be the price of the soda alone, which we find by substituting h back into one of the equations, resulting in s = $2.