How many of each ticket were sold in one day? Child tickets - $\$7$ 

Adult Tickets - $\$10$

Senior Tickets - $\$5$

Day one sold $678$ tickets for $\$5,812$

Day two sold $535$ tickets for $\$4,541$

How many of each ticket were sold on day one and two?

If someone could even just explain how to set it up or what method to use I would appreciate it a lot. Thank you!
Let $x$ be the number of child tickets sold, $y$ adults, and $z$ seniors. Then we know for day one 
$$
\$x + \$y + \$z = \$ 5812
$$ 
and 
$$
x + y + z = 678.
$$ 
What are the dollar amounts associated with $x,y$, and $z$? What does day two look like?
Child tickets - $7

Adult Tickets - $10

Senior Tickets - $5

A is the number of Adult tickets, C is the number of Child tickets, and S is the number of Senior tickets in what follows below. 

A1 + C1 + S1 = 678 (Day 1)

A2 + C2 + S2 = 535 (Day 2)

Solution: 

Day one: Sold tickets 678 for $5,812

A1 = 420, C1 = 161, S1 = 97 (Day 1)

Day two: Sold tickets 535 for $4,541

A2 = 215, C2 = 38, S2 = 425 (Day 2)

It must be remembered that other answers are also possible.