A suitable graphing caculator gives the result
X = 15 25/39
Y = 7 6/13
Z = 4 4/39
_____
If you want to solve this by hand, you can add 3 times the first equation to the second to get
... 5X -2Z = 70
and 5 times the first equation to the third to get
... 8X -11Z = 80
Then Cramer's Rule tells you the solutions for X and Z are
... X = (-160+770)/(-16+55) = 610/39
... Z = (560 -400)/39 = 160/39
From there, you can substitute into any of the equations to find Y. The first one might be most convenient.
... Y = 19 -X +Z = 19 -(15 25/39) +(4 4/39) = 8 -21/39 = 7 6/13