a +    c = 124,    (1)    ("attendie" equation)
6a + 3.5c = 679.    (2)    {"value" equation)

Here, as you can guess, "a" stands for the number of adults and "c" stands for the number of children.


From (1), express a = 124-c, and substitute it into (2). You will get

6*(124-c) + 3.5c = 679.


This is a single equation for c.

If the data is correct, it has an integer solution.

You can solve it on your own.
Let it be an exercise to empower you brain.

Let "c" be the number of children. 
Then the number of adult is (124-c).

Then you can write a "value equation

6*(124-c) + 3.5c = 679.


Notice that it coincides with the equation of the Solution 1.


Again, you can complete the solution on your own.