That's a really weird way to pose a math problem. Hope I can clarify this a bit more:

If you express the answer as a ratio, you basically can set one of the variables to any value. For the sake of argument, let z = 1.

If 6y = 5z, then y = ⅚z. If z = 1, then y = 5/6.

Same with 10x = 7z -> x = 7/10

The right ratio would be

7/10 : 5/6 : 1

You can multiple by 30 to have an integer answer

21 : 25 : 30

Edit: you can also just invert the numbers you have. If 50x = 42y = 35z then the ratio is 1/50 : 1/42 : 1/35, multiplying everything by 1050 it's still 21 : 25 : 30