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

First, let's rewrite the equations in standard form:

1) X1 + 2X2 - 2X3 = -16
2) 3X1 + 4X2 - 5X3 = -13
3) 2X1 + 0X2 + 3X3 = 13

Now, let's use equations 1 and 2 to eliminate X1:

1) X1 + 2X2 - 2X3 = -16
2) 3X1 + 4X2 - 5X3 = -13

Multiply equation 1 by 3 and equation 2 by 1:

3) 3X1 + 6X2 - 6X3 = -48
4) 3X1 + 4X2 - 5X3 = -13

Subtract equation 4 from equation 3:

2X2 + X3 = -35

Next, let's use equations 1 and 3 to eliminate X1:

1) X1 + 2X2 - 2X3 = -16
3) 2X1 + 0X2 + 3X3 = 13

Multiply equation 1 by 2 and equation 3 by -1:

5) 2X1 + 4X2 - 4X3 = -32
6) -2X1 + 0X2 - 3X3 = -13

Sum equation 5 and equation 6:

4X2 - 7X3 = -45

Now we have a system of two equations:

1) 2X2 + X3 = -35
2) 4X2 - 7X3 = -45

We can solve this system of equations using substitution or elimination to find the values of X2 and X3.