<PRE><B>Question 1192827</B>
&lt;br&gt;
3x + 2y + 4z = 12
x + 5y + 2z = -9
7x + 2y + z = 5&lt;br&gt;
I have to admit that this method is new to me; I have never run across it before.  Thanks for submitting a question that lets me learn something new.&lt;br&gt;
After viewing a couple of online videos, it appears the method is well defined.&lt;br&gt;
(1) Rearrange the equations so that the system is in &quot;diagonally dominant&quot; form:&lt;br&gt;
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NOTE: See the response from tutor @Edwin for a description of &quot;diagonally dominant form&quot;....&lt;br&gt;
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7x + 2y + z = 5
x + 5y + 2z = -9
3x + 2y + 4z = 12&lt;br&gt;
(2) Solve the first equation for x; the second for y; and the third for z:&lt;br&gt;
[1] x=(5-2y-z)/7
[2] y=(-9-x-2z)/5
[3] z=(12-3x-2y)/4&lt;br&gt;
(3) Choose random starting values x(0), y(0), and z(0) for the three variables.&lt;br&gt;
(4) Use the most recent values of y and z in [1] to compute the next iteration of x;
(5) Use the most recent values of x and z in [2] to compute the next iteration of y;
(6) Use the most recent values of x and y in [3] to compute the next iteration of z;
(7) Repeat (4) to (6) until the desired number of iterations have been made.&lt;br&gt;
The calculations are impossible by hand; and they are extremely awkward using a calculator.  However, they are quite easy to perform using a spreadsheet program.&lt;br&gt;
Here is a description of how to do that.&lt;br&gt;
Make column A the iteration number: A1=0, A2=1, ..., A6=5 (since you are doing 5 iterations).
Columns B, C, and D will be the iterated values of x, y, and z.  Put your initial random guesses (iteration 0) in row 1: B1=0; C1=0; D1=0. (you can use any numbers there....)
Cell B2 will contain the first iteration of x, using equation [1] and the most recent values of variables y and z:  B2=(5-2*C1-D1)/7.
Cell C2 will contain the first iteration of y, using equation [2] and the most recent values of variables x and z:  C2=(-9-B2-2*C1)/5.
Cell D2 will contain the first iteration of z, using equation [3] and the most recent values of variables x and y:  D2=(12-3*B2-2*C2)/4&lt;br&gt;
Now for the beauty of excel....&lt;br&gt;
To compute the remaining iterations 2 through 5, simply copy the formulas in cells B2 to D2 down into rows 3 through 6 by highlighting cells B2 to D2 and using ctrl-D to drag those formulas into the other rows.&lt;br&gt;
I don&#39;t know how to copy the results of my excel file here, so I will recreate it....&lt;br&gt;&lt;pre&gt;

 iteration   x          y          z
     0       0          0          0
     1    0.714286  -1.94286   3.435714
     2    0.778571  -3.33      4.081071
     3    1.082704  -3.64897   4.012457
     4    1.18364   -3.64171   3.933125
     5    1.192899  -3.61183   3.91124&lt;/pre&gt;
So the solution after 5 iterations is&lt;br&gt;
(1.192899, -3.61183, 3.91124)&lt;br&gt;
The exact solution is (122/103, -371/103, 403/103)&lt;br&gt;
which to several decimal places is&lt;br&gt;
(1.184466, -3.60194, 3.912621)&lt;br&gt;
Note if you do this exercise in excel on your own and continue for about 14 iterations, the solutions will be exact to 5 or 6 decimal places.&lt;br&gt;

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