To solve this system of equations, we can use the method of substitution or elimination. Let's use substitution in this case.

From the first equation, we have:
5x + y - 3z = -2
y = -5x + 3z - 2

Now substitute y into the second and third equations:

4x + 3(-5x + 3z - 2) + 2z = 16
4x - 15x + 9z - 6 + 2z = 16
-11x + 11z = 22
x - z = -2 (1)

2x - 3(-5x + 3z - 2) + z = 17
2x + 15x - 9z + 6 + z = 17
17x - 8z = 11
17x = 8z + 11
x = (8/17)z + 11/17 (2)

Substitute (2) into equation (1):

(8/17)z + 2 - z = -2
(8/17)z - z = -2 - 2
-(9/17)z = -4
z = -4*(-17/9)
z = 68/9

Now, substitute z back into equation (2) to solve for x:

x = (8/17)*(68/9) + 11/17
x = 32/3

Finally, substitute x and z back into the equation y = -5x + 3z - 2 to solve for y:

y = -5(32/3) + 3(68/9) - 2
y = -160/3 + 204/9 - 2
y = -160/3 + 68/3 - 2
y = -92/3 - 6
y = -104/3

Therefore, the solution to the system of equations is:
x = 32/3, y = -104/3, z = 68/9.