Eliminating y right away is easiest. Add -3(equation 1) to equation 2, and add 3(equation 3) to equation 2:
-3(2x + y + 3z) + (-3x + 3y - 2z) = -3(3) + (-8)
-6x - 3y - 9z - 3x + 3y - 2z = -9 - 8
-9x - 11z = -17
9x + 11z = 17 [equation 1]
3(5x - y + 5z) + (-3x + 3y - 2z) = 3(13) + (-8)
15x - 3y + 15z - 3x + 3y - 2z = 39 - 8
12x + 13z = 31 [equation 2]
Eliminating x next is easier than eliminating z. Add -4(equation 1) to 3(equation 2):
-4(9x + 11z) + 3(12x + 13z) = -4(17) + 3(31)
-36x - 44z + 36x + 39z = -68 + 93
-5z = 25
z = 25/(-5)
z = -5
Plug this into either of the previous equations and solve for x :
9x + 11z = 17
9x + 11(-5) = 17
9x - 55 = 17
9x = 72
x = 72/9
x = 8
Plug x and z into any of the 3 original equations and solve for y :
2x + y + 3z = 3
2(8) + y + 3(-5) = 3
16 + y - 15 = 3
1 + y = 3
y = 2