System of Linear Equations Online

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.

What is System of Linear Equations?

System of Linear Equations is a free, browser-based tool that helps you get the job done quickly. Solve linear systems with RREF, Gauss-Jacobi, and Gauss-Seidel methods using full step-by-step tables. It's built for speed and privacy: Everything runs locally in your browser — your data is never uploaded to a server. No sign-up, no installs, and no daily limits.

How to use System of Linear Equations

1Paste or drop your data into the workspace.
2Adjust the available options to suit your data.
3Copy or download the result — it never leaves your browser.

Features

Fast, focused, and easy to use
100% client-side — no upload, fully private
Free with no account, no watermark, and no daily limit
Works on large inputs, limited only by your device's memory
Load a sample, then copy, download, or share your result

Common uses

Quick one-off data jobs without opening a heavy app
Preparing data before importing it elsewhere
Sharing a reproducible result with a teammate via a link

Used by

Students
Finance teams
Operations teams
Analysts

Examples

Example 1

Example input

Goal: calculate a system of linear equations result
Values: sample amount, rate, and time period

Example output

The calculator shows the final value, intermediate figures, and any percentage or difference used in the result.

Example 2

Simple case

A small input for System of Linear Equations

Immediate result

A focused result you can review before copying or downloading.

Example 3

Custom case

Input with changed settings or options

Adjusted result

The output updates to match your selected options.

Example 4

Review case

A value you want to double-check before using elsewhere

Reviewable output

The result gives you a safer handoff into the next task.

Example 5

Workflow case

Input prepared by another DataXForge tool

Next-step result

The output can be copied, downloaded, or used in another browser workflow.

Common errors

Input is copied from a source with hidden whitespace or encoding issues.
The result is not reviewed before being reused elsewhere.
The wrong options are selected for the target workflow.
The original data is not kept for comparison.

How it works

1Accepts the input for System of Linear Equations.
2Runs the operation in the browser.
3Returns a result you can copy, download, or continue using.

Recommended workflow

1
Prepare the input
Use this tool as the focused step in a larger browser-based workflow.
2
Run System of Linear Equations
Review the preview, copy or download the result, and keep everything local in your browser.
3
Next recommended tool
Curve Fitting Calculator: Least-squares line, parabola, exponential, geometric, and power curve fitting with full equation tables.
Open tool
4
Then continue with
Interpolation Calculator: Step-by-step Newton forward/backward, divided difference & Lagrange interpolation with full tables.
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5
Then continue with
Scientific Calculator: Trig, log, exp, powers — expression evaluator.
Open tool

Frequently asked questions

How do I use System of Linear Equations?: Paste or drop your data into the workspace, adjust the options to fit your data, then copy or download the result. Solve linear systems with RREF, Gauss-Jacobi, and Gauss-Seidel methods using full step-by-step tables.
Is my data uploaded or stored anywhere?: No. Everything runs locally in your browser — your data is never uploaded to a server. Nothing is sent to a server, so it's safe to use with private or sensitive data.
Is System of Linear Equations free?: Yes — it's completely free with no account, no sign-up, no watermark, and no daily usage limits.
What's the maximum file size?: There's no hard limit imposed by the tool. Because processing happens in your browser, the practical ceiling is your device's available memory — most everyday files work instantly.
Do I need to install anything?: No installs and no extensions — it runs entirely in your web browser and even works offline once the page has loaded.

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

What is System of Linear Equations?

Examples

Example 1

Example input

Goal: calculate a system of linear equations result
Values: sample amount, rate, and time period

Example output

The calculator shows the final value, intermediate figures, and any percentage or difference used in the result.

Example 2

Simple case

A small input for System of Linear Equations

Immediate result

A focused result you can review before copying or downloading.

Example 3

Custom case

Input with changed settings or options

Adjusted result

The output updates to match your selected options.

Example 4

Review case

A value you want to double-check before using elsewhere

Reviewable output

The result gives you a safer handoff into the next task.

Example 5

Workflow case

Input prepared by another DataXForge tool

Next-step result

The output can be copied, downloaded, or used in another browser workflow.

Recommended workflow

Prepare the input

Use this tool as the focused step in a larger browser-based workflow.

Run System of Linear Equations

Review the preview, copy or download the result, and keep everything local in your browser.

Next recommended tool

Curve Fitting Calculator: Least-squares line, parabola, exponential, geometric, and power curve fitting with full equation tables.

Open tool

Then continue with

Interpolation Calculator: Step-by-step Newton forward/backward, divided difference & Lagrange interpolation with full tables.

Open tool

Then continue with

Scientific Calculator: Trig, log, exp, powers — expression evaluator.

Open tool

Frequently asked questions

How do I use System of Linear Equations?

Paste or drop your data into the workspace, adjust the options to fit your data, then copy or download the result. Solve linear systems with RREF, Gauss-Jacobi, and Gauss-Seidel methods using full step-by-step tables.

Is my data uploaded or stored anywhere?

No. Everything runs locally in your browser — your data is never uploaded to a server. Nothing is sent to a server, so it's safe to use with private or sensitive data.

Is System of Linear Equations free?

Yes — it's completely free with no account, no sign-up, no watermark, and no daily usage limits.

What's the maximum file size?

There's no hard limit imposed by the tool. Because processing happens in your browser, the practical ceiling is your device's available memory — most everyday files work instantly.

Do I need to install anything?

No installs and no extensions — it runs entirely in your web browser and even works offline once the page has loaded.