Somewhat simpler is to note that given (x+10)/5 = y-7 = (z+3)/4, this can be written differently. Let the common ratio be equal to A.
Then (x+10)/5 = A --> x = 10+5A, and similarly y = 7+A, z = 3+4A.
Thus (x,y,z) = (10,7,3) + A(5,1,4) = (10,7,3)+ A'(10,2,8) (where A' = A/2)
It is now...