The total cost includes the setup cost + per hour cost.
Let, x represents the time and y represents the total cost.
.... (1)
Let Y represent the cost for B's DJ service.
.... (2)
a. Graphing both the equations give the intersection point (solution) as (4,500)
It means for 4 hours, the total cost of both the DJ's is same at $500.
b. Substitution Method.
Plug y = 125x in equation (1)
![125x= 200+75x](https://tex.z-dn.net/?f=125x%3D%20200%2B75x)
![125x-75x=200](https://tex.z-dn.net/?f=125x-75x%3D200)
![50x=200](https://tex.z-dn.net/?f=50x%3D200)
![x=4](https://tex.z-dn.net/?f=x%3D4)
![y=125\times 4 =500](https://tex.z-dn.net/?f=y%3D125%5Ctimes%204%20%3D500)
Hence, the total cost for 4 hours in both the service is $500.
c. Addition method.
Multiply equation (2) by -1
.... Equation (3).
Add euation (1) and equation (3)
![y+(-y) = 200+75x-125x](https://tex.z-dn.net/?f=y%2B%28-y%29%20%3D%20200%2B75x-125x)
![0=200-50x](https://tex.z-dn.net/?f=0%3D200-50x)
![50x=200, x=4](https://tex.z-dn.net/?f=50x%3D200%2C%20x%3D4)
![y=500](https://tex.z-dn.net/?f=y%3D500)
Solution: For x = 4, y = 500 ( same as the previous two methods)
For x = 2, equation (1)
![y=200+(75\times 2) = 350](https://tex.z-dn.net/?f=y%3D200%2B%2875%5Ctimes%202%29%20%3D%20350)
For x = 2, equation (2)
![y= 125\times 2 =250](https://tex.z-dn.net/?f=y%3D%20125%5Ctimes%202%20%3D250)
Hence, for 2 hours, DJ B would be a better choice since it would charge $250.
For x = 6 hours, equation (1)
![y=200+(75\times 6) =650](https://tex.z-dn.net/?f=y%3D200%2B%2875%5Ctimes%206%29%20%3D650)
For x= 6, equation (2)
![y=125\times 6=750](https://tex.z-dn.net/?f=y%3D125%5Ctimes%206%3D750)
For 6 hours, DJ A would be better option since it charges $650 for 6 hours.