Variables: let "x" be the numbers of adult tickets, and
           let "y" be the numbers of student tickets.

The system is:
 x +    y = 100,       (1)   ("100 tickets were sold") and
4x + 2.5y = 355.       (2)   ("100 tickets were sold for $355.00")

To solve the system, express "x" from (1): x = 100 - y, and substitute it into (2). You will get a single equation for y:

4*(100 - y) + 2.5y = 355.

Simplify and solve it:

400 - 4y + 2.5y = 355,   or

-1.5y = 355 - 400,

-1.5y = -45,

y =  = 30.

So, we just found y, the number of student tickets.

Then  x = 100 - y = 100 - 30 = 70. It is the number of adult tickets.

Check: 70*4 + 30*2.5 = 280 + 75 = 355.  Correct !

Answer.  70 adult and 30 student tickets were sold.