Translate the problem into a pair of linear equation in two vaiables. Solve the equations using either?

number of adults =x
number of students=y = number of tickets Ron sold

2y + 4.30x = 1175.50

x + y = 364

use the 2 equations ====> 2 ( 364 - x ) + 4.30 x = 1175.50  ==> x = 192  , y= 172

x = number of tickets sold by Ron
y = number of tickets sold by Kathy

x + y = 364.......................(i)
2x +4.30 y = 1175.50.............(ii)
Multiply (i) by 2
2x +2y = 728...................(iii)
subtract (iii) from (ii)
2.30y = 447.50
        y = 104.06976744186047
           = 104......................(iv)
Therefore x = 364 - 104 = 260...................(v)
Hence Ron sold = 260 tickets...................... ..........Ans
Please ckeck, may be printing mistake
If adults ticket is $4.50, then y = 179
and x = 364 - 179 = 185 tickets sold by Ron............Ans

s+a=364
2s+4.3a=1175.50

a=364-s

2s+4.3(364-s)=1175.50
2s+1565.2-4.3s=1175.50
-2.3s+1565.20=1175.50
-2.3s=1175.50-1565.20
-2.3s=-389.7
s=169.43  169*2=338  s= 169 tickets

1175.50-338=837.50 /4.3=194.76  195 tickets

195*4.3=837.50 169*2=338=  364 tickets for 1175.50

None of the first 3 answers is correct. Perhaps one of the numbers in the statement was mistyped. There is no consistent answer where Ron and Kathy each sell a whole number of tickets totaling 364 and come up with a revenue of $1175.5.
for x = Kathy's tickets and y = Ron's tickets
x + y = 364
4.3x + 2y = 1175.5
substituting from the first eqtn into the 2nd for y you get 4.3x + 2(364 -x) = 1175.5
2.3x = 447.5 , x = 194.57 and that is inconsistent because it should be a whole number.
Spider Monkey came the closest, but he rounded off instead of realizing the problem is inconsistent.