Method

Decimal places

Iterations

Tolerance

Initial guess

Reorder rowsfor diagonal dominance

Augmented matrix

3 equations · 3 variables

Eq	x	y	z	b	Action
R1
R2
R3

20 2 6 | 28

Paste or edit equations / augmented matrixCSV

Matrix form

Coefficient matrix A

20	2	6
1	20	9
2	-7	-20

Unknowns X

Constants b

-23

-57

Strict diagonal dominance

Row	Diagonal	Others	Check
R1 (x)	\|20\| = 20	8	strictly dominant
R2 (y)	\|20\| = 20	10	strictly dominant
R3 (z)	\|-20\| = 20	9	strictly dominant

Gauss-Jacobi formulas

x = (28 - 2*y - 6*z) / 20

y = (-23 - 1*x - 9*z) / 20

z = (-57 - 2*x + 7*y) / -20

Stopping rule

After each iteration compute max change = max(|new value - previous value|).

Stop rule: stop when max change <= tolerance 0.0005, or stop at the maximum 9 iterations.

Stopped at iteration 9 because the maximum iteration limit (9) was reached; last max change = 0.0009.

The approximate answer is taken from the last row of the iteration table. If tolerance, maximum iterations, initial guess, or method changes, the stopping iteration and final answer may change.

Iteration table

k	x	y	z	max change
1	1.4	-1.15	2.85	2.85
2	0.66	-2.5025	3.3925	1.3525
3	0.6325	-2.7096	3.7919	0.3994
4	0.5334	-2.888	3.8616	0.1783
5	0.5303	-2.9144	3.9141	0.0525
6	0.5172	-2.9379	3.9231	0.0235
7	0.5169	-2.9412	3.93	0.0069
8	0.5151	-2.9443	3.9311	0.0031
9	0.5151	-2.9448	3.932	0.0009

Highlighted row is the stopping row used for the approximate answer.

Stopped at iteration 9 because the maximum iteration limit (9) was reached; last max change = 0.0009.

Substitution steps

Iteration 1

x^(1) = (28 - 2*(0) - 6*(0)) / 20 = 1.4

y^(1) = (-23 - 1*(0) - 9*(0)) / 20 = -1.15

z^(1) = (-57 - 2*(0) + 7*(0)) / -20 = 2.85

Iteration 2

x^(2) = (28 - 2*(-1.15) - 6*(2.85)) / 20 = 0.66

y^(2) = (-23 - 1*(1.4) - 9*(2.85)) / 20 = -2.5025

z^(2) = (-57 - 2*(1.4) + 7*(-1.15)) / -20 = 3.3925

Iteration 3

x^(3) = (28 - 2*(-2.5025) - 6*(3.3925)) / 20 = 0.6325

y^(3) = (-23 - 1*(0.66) - 9*(3.3925)) / 20 = -2.7096

z^(3) = (-57 - 2*(0.66) + 7*(-2.5025)) / -20 = 3.7919

Iteration 4

x^(4) = (28 - 2*(-2.7096) - 6*(3.7919)) / 20 = 0.5334

y^(4) = (-23 - 1*(0.6325) - 9*(3.7919)) / 20 = -2.888

z^(4) = (-57 - 2*(0.6325) + 7*(-2.7096)) / -20 = 3.8616

Iteration 5

x^(5) = (28 - 2*(-2.888) - 6*(3.8616)) / 20 = 0.5303

y^(5) = (-23 - 1*(0.5334) - 9*(3.8616)) / 20 = -2.9144

z^(5) = (-57 - 2*(0.5334) + 7*(-2.888)) / -20 = 3.9141

Iteration 6

x^(6) = (28 - 2*(-2.9144) - 6*(3.9141)) / 20 = 0.5172

y^(6) = (-23 - 1*(0.5303) - 9*(3.9141)) / 20 = -2.9379

z^(6) = (-57 - 2*(0.5303) + 7*(-2.9144)) / -20 = 3.9231

Iteration 7

x^(7) = (28 - 2*(-2.9379) - 6*(3.9231)) / 20 = 0.5169

y^(7) = (-23 - 1*(0.5172) - 9*(3.9231)) / 20 = -2.9412

z^(7) = (-57 - 2*(0.5172) + 7*(-2.9379)) / -20 = 3.93

Iteration 8

x^(8) = (28 - 2*(-2.9412) - 6*(3.93)) / 20 = 0.5151

y^(8) = (-23 - 1*(0.5169) - 9*(3.93)) / 20 = -2.9443

z^(8) = (-57 - 2*(0.5169) + 7*(-2.9412)) / -20 = 3.9311

Iteration 9

x^(9) = (28 - 2*(-2.9443) - 6*(3.9311)) / 20 = 0.5151

y^(9) = (-23 - 1*(0.5151) - 9*(3.9311)) / 20 = -2.9448

z^(9) = (-57 - 2*(0.5151) + 7*(-2.9443)) / -20 = 3.932

Residual check

Equation 1: Ax - b = 0.0046

Equation 2: Ax - b = 0.0081

Equation 3: Ax - b = 0.0029

Approximate solution

x = 0.5151

y = -2.9448

z = 3.932

Taken from the last iteration row. Changing tolerance, maximum iterations, initial guess, or method can change this approximate result.

Tip: use samples, upload, copy, download, and send-to actions inside the workspace where available.

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Action

Row

Diagonal

Others

Check

R1 (x)

|20| = 20

strictly dominant

R2 (y)

|20| = 20

strictly dominant

R3 (z)

|-20| = 20

strictly dominant

max change

1.4

-1.15

2.85

0.66

-2.5025

3.3925

1.3525

0.6325

-2.7096

3.7919

0.3994

0.5334

-2.888

3.8616

0.1783

0.5303

-2.9144

3.9141

0.0525

0.5172

-2.9379

3.9231

0.0235

0.5169

-2.9412

3.93

0.0069

0.5151

-2.9443

3.9311

0.0031

0.5151

-2.9448

3.932

0.0009

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