Let A = number of Adults who attended
Let S = number of Students who attended
Let C = number of senior Citizens who 
attended

A total of 434 people attended a community 
theatre performance.

A + S + C = 434

The admission prices were $9.00 for 
adults, $7.50 for students, and $8.00 for 
senior citizens. The ticket sales totaled 
$3712. 

9A + 7.5S + 8C = 3712

At the theatre's next board meeting, the 
finance manager proposed that for the next 
play they raise prices to $11.00 for adults, 
$8.50 for students, and $8.50 for senior 
citizens. He said that if the exact same 
number of people attend the next performance, 
the ticket sales would be $4374. 

11A + 8.5S + 8.5C = 4374

So we have this system of three 
equations in three unknowns.

  A +    S +    C =  434
 9A + 7.5S +   8C = 3712
11A + 8.5S + 8.5C = 4374

You can simplify the 2nd and 3rd 
equations:

Simplifying the 2nd equation:

   9A + 7.5S + 8C = 3712

First clear of decimals by multiplying 
through by 10

  90A + 75S + 80C = 37120

Now divide every term through by 5

   18A + 15S + 16C = 7424

Simplifying the 3rd equation:

11A + 8.5S + 8.5C = 4374

First clear of decimals by multiplying 
through by 10

110A + 85S + 85C = 43740

Now divide every term through by 5

22A + 17S + 17C = 8748

Now the system is

  A +   S +   C =  434
18A + 15S + 16C = 7424
22A + 17S + 17C = 8748

That's a little easier to solve.
Can you solve it? If not post
again asking how to.

Answer:
A = 274 S = 68, C = 92,
that is, 274 adults, 68 students, 
and 92 senior citizens attended 
the play.

Edwin