11x+7y=180 is a simplification of the given equation.
There are an infinite number of solutions for x and y in the realm of real numbers. The graph of the equation as a line represents the relationship between x and y.
But if x and y are restricted to integers, then y=(180-11x)/7=25-x-(4x-5)/7.
If y is an integer, 4x-5 must be exactly divisible by 7. When x=3, 4x-5=7, which is divisible by 7. Similarly, when x=10, 17, 24,... 4x-5=35, 63, 91,...
Corresponding values of y are (180-11x)/7=21, 10, -1, -12. These values form an arithmetic series with common difference -11.
The constant 180 suggests geometry and supplementary angles 11x and 7y, so we would restrict these angles to the interval [0,180]. The only candidates are when x=3 or 10 and y=21 or 10, that is, angles 33 and 147, 110 and 70 degrees.
Returning to the original question we can substitute for x and y:
15+63+18+84=180 or
50+30+60+40=180.