⅓x+¼y=12 only relates x and y and there are many solutions (an infinite number) for x and y.
The solutions are limited if x and y are natural numbers.
4x+3y=144 is another way of writing this. So, y=(144-4x)/3=48-(4/3)x.
When x is any multiple of 3, y is a an integer≥0 for:
Examples: x=3, y=44; x=6, y=40; x=9, y=36; ...; x=18, y=24; x=21, y=20; ...; x=27, y=12; ...; x=36, y=0.