Nola was selling tickets at the high school dance. At the end of the evening, she picked up the cash box and noticed a dollar lying on the floor next to it. She said, I wonder whether the dollar belongs inside the cash box or not. The price of tickets for the dance was 1 ticket for $5 (for individuals) or 2 tickets for $8 (for couples). She looked inside the cash box and found $200 and ticket stubs for the 47 students in attendance. Does the dollar belong inside the cash box or not? (justify/defend)

yes

Step-by-step explanation:

s + 2c = 47

5s+8c = 200

which one checks as in the other solution does not have a solution in non-negative integers. On the other hand, the analogous system

s + 2c = 47

5s+8c = 201

with solution s=13 and c=17. So, as in the other solution, we conclude that the extra dollar belongs in the cash box, with 13 singles and 17 couples attending the dance.

check above link.

graphically?   well, you graph them both, the solutions is where they both intersect each other, since there are two solutions, meaning they will meet twice, or namely the linear will run into the quadratic curve twice, and where they meet, those x,y coordinates are the solutions to the system.