Final answer:
The shop needs 17 small wheels and 58 large wheels, which are calculated by forming and solving a system of equations based on the relationship between the number of small and large wheels and the total needed.
Step-by-step explanation:
To solve the problem, we need to set up and solve a system of equations based on the given information. Let's denote the number of small wheels as x and the number of large wheels as y. From the problem, we can establish the following relationships:
The number of large wheels is seven more than three times the number of small wheels: y = 3x + 7.
The total number of wheels needed is 75: x + y = 75.
Now, we will solve the system of equations:
Replace y in the second equation with the expression from the first equation to get: x + (3x + 7) = 75.
Combine like terms: 4x + 7 = 75.
Subtract 7 from both sides: 4x = 68.
Divide by 4: x = 17.
This tells us that there are 17 small wheels.
Now, to find the number of large wheels, substitute x back into the first equation: y = 3(17) + 7.
Calculate y: y = 51 + 7 = 58.
This tells us that there are 58 large wheels.
Therefore, the shop needs 17 small wheels and 58 large wheels.