First, rearrange the equations for x, y, z = expression in other variables:
x=-0.2y-0.1z+0.9; y=-0.1x+0.1z-2.2; z=0.2x-0.3y+2.2.
These are the iteration formulas to be used in the method.
We need to start with the initial values of 0 for all three variables. We work through the equations in order for x, y and z.
Iteration 1:
With y=z=0, we get x=0.9 (because the terms in y and z drop out). So we have a new value for x to replace 0.
Now for y: y=-0.1*0.9-2.2 (because z term drops out). We get y=-2.29.
Then z: 0.2*0.9+0.3*2.29+2.2=3.067.
That completes iteration 1.
Iteration 2:
x=0.2*2.29-0.1*3.067+0.9=1.0513;
y=-0.1*1.0513+0.1*3.067-2.2=-1.78817;
z=0.2*1.0513+0.3*1.78817+2.2=2.946711.
Iteration 3:
x=0.2*1.78817-0.1*2.946711+0.9=0.9629629;
y=-0.1*0.9629629+0.1*2.946711-2.2=-1.80903261;
z=0.2*0.9629629+0.3*1.80903261+2.2=2.935302363.
The iterations seem to be converging on x=1, y=-2, z=3, so at this point we can apply these values to the original equations:
10-4+3=9; 2-40-6=-44; -2-6+30=22. So all the equations are satisfied, so (1,-2,3) is the solution.
In practice, it is probably unnecessary to compute iterative values to complete accuracy. Two decimal places would probably be sufficient and lead more quickly to the solution. When one of the values remains the same on successive iterations, convergence is complete for that variable. The process continues until all variables are stable.