I take it from the title that this is really supposed to be
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The first thing that tells us is that right hand side is an integer and so (19x+ 16)/10 must be an integer: (19x+ 16)/10= n so 19x+ 16= 10n for some integer n. From that 19x= 10n-16 and x= (10n- 16)/19.
Putting that into the right side, (4x+ 7)/3= (40n+ 69)/57 and that must have "floor" n. That is, it must be equal to n+ where is between 0 and 1.
(40n+ 69)/57= n+ so or and, finally, .
Since the largest can be is 1, n cannot be smaller than (69-57)/17= 12/17= 0.705 .... Since the smallest can be is 0, n cannot be larger than 69/17= 4.04. Since n must be a constant n must be 1, 2, 3, or 4.
If n= 1, then 19x+ 16= 10 and x= -6/19.
If n= 2, then 19x+ 16= 20 and x= 4/19.
If n= 3, then 19x+ 16= 30 and x= 14/19.
If n= 4, then 19x+ 16= 40 and x= 24/19.
I will leave it to you to check that all those do, in fact, satisfy the original equation.
Blast, too slow!! Ah, well, ladies first.