semkuSolving systems of equations calculator
Solve two or three linear equations.
Linear system and elimination steps
Original equations
- Equation 1: 3x + 2y = 7
- Equation 2: x − 4y = 5
Page: https://semku.com/solving-systems-of-equations-calculator/en
Show steps
Original augmented matrix
| x | y | right-hand side |
|---|---|---|
| 3 | 2 | 7 |
| 1 | −4 | 5 |
Divide row 1 by 3: R1 ← R1 ÷ 3.
Augmented matrix after step 1
| x | y | right-hand side |
|---|---|---|
| 1 | 2/3 | 7/3 |
| 1 | −4 | 5 |
Eliminate x from row 2: R2 ← R2 − R1.
Augmented matrix after step 2
| x | y | right-hand side |
|---|---|---|
| 1 | 2/3 | 7/3 |
| 0 | −14/3 | 8/3 |
Divide row 2 by −14/3: R2 ← R2 ÷ (−14/3).
Augmented matrix after step 3
| x | y | right-hand side |
|---|---|---|
| 1 | 2/3 | 7/3 |
| 0 | 1 | −4/7 |
Eliminate y from row 1: R1 ← R1 − (2/3)·R2.
Reduced augmented matrix
| x | y | right-hand side |
|---|---|---|
| 1 | 0 | 19/7 |
| 0 | 1 | −4/7 |
Check my answer
Calculation rules
The calculator solves square systems of two or three linear equations by exact Gauss–Jordan elimination: choose the first nonzero pivot, make that pivot 1, then eliminate its column above and below. Each row operation and resulting augmented matrix is shown.
A row with zero coefficients and a nonzero right-hand side means no solution. Otherwise, a pivot for every variable means one solution; fewer pivots mean infinitely many solutions, with free variables represented by parameters.
Parameters in a general solution can be any real numbers. Substituting them into the original equations verifies the family of solutions. Substitution and other exact methods give the same solution set.
Fractions remain exact throughout elimination. Decimal display rounds once to the selected precision, with halfway values rounded away from zero; = denotes equality and ≈ denotes approximation. A fraction too long for display is shown approximately without changing the calculation.
Nonlinear terms such as x² or xy, inequalities, four or more variables and unequal numbers of equations and variables are outside this calculator's scope. The prefilled equations are examples. Print steps includes all elimination steps, even when Show steps is closed.
Checking first substitutes your values exactly. For a unique solution, unequal decimal entries can match rounding at each entry's submitted precision; this does not identify the cause of a difference. Infinitely many solutions are checked by exact substitution only. A system with no solution cannot accept an exact answer.
Frequently asked questions
- What does 0 = 1 mean?
- Elimination produced a contradiction, so no values can satisfy all the original equations.
- What does t mean in the answer?
- It is a free parameter; each real value gives a solution in the displayed family.
- Can I solve nonlinear or four-variable systems?
- This calculator handles only two or three linear equations in the same number of variables.
- What does Print steps include?
- It includes the original equations, result state, every row operation and matrix, the solution and its check; long work can span pages.
Privacy on this page
Calculations run in your browser. The values you enter are not sent to the server or saved in browser storage. Reloading the page returns the inputs to their initial state.
To understand how the page is used and to improve the service, this page sends semku’s own usage statistics and automatic error reports. They go only to the dedicated endpoint under semku.com/api/telemetry/v1/. They are not used for advertising or third-party tracking.
The usage statistics include the stage of use (such as starting an operation or requesting a copy), the time it happened and fixed identifiers that distinguish the page and feature. They do not include the values you enter, the results shown or the text you copy. Inputs are not sent as hashes or as size categories either. The full URL and search terms are not sent.
Operations and sessions are distinguished by randomly generated identifiers. In the browser these exist only in temporary memory and are not written to cookies or browser storage. No identification based on cookies or combined device characteristics is used.
Error reports contain the error type, code and time, and, where available, a registered program-source identifier with the line and column number. The error message text and stack traces are not sent. Usage statistics and error reports also carry the site identifier, the published build and a random record identifier. For access records such as IP addresses, browser signals (User-Agent Client Hints), region and language, see the paragraph below.
The feedback form sends the rating you chose and the text you typed in the free-text field when you press the send button. A random ID that distinguishes the submission, the page, site and published build, and the type of request are also sent. There are no fields for your name, email address or phone number, and the values you enter are not attached automatically. Whatever you write in the free-text field is sent as is. Please do not enter personal data, passwords, authentication keys or payment details.
To improve the service, we record the country and state/province-level region at the time of your visit (an estimate based on the connecting network), which site you came from including the page path, and one display-language setting announced by your browser (for example, en-US). Location is kept to country and region only; no city or precise position is recorded. When you use the service, connection records, including your IP address, are automatically generated and retained to maintain service security and identify abnormal access. Cloudflare is the data processor for this site. Collection of this item is enabled only after this notice is published; we do not collect it retrospectively for use before collection is enabled. Query strings such as search terms are not recorded, and neither is the full list of preferred languages or the raw request headers. To identify bots, we may collect and use signals your browser reports about its type and version, along with an indication of whether automation tools are being used. These browser signals are estimates and do not conclusively identify your browser's type or version or whether automation tools are being used. Collection of this information is currently disabled and will only be enabled after this notice is published. We do not collect it retrospectively for use before collection is enabled. The raw User-Agent request header is not recorded. These records are per visit only and use no identifier that links one visit to another. Recording starts only with visits after this notice is published and is not applied retroactively to earlier visits. The region and the referring site have been recorded since 2026-09-20; the display language since 2026-09-24. Browser signals are not recorded yet.
Separately, when you visit the site, the hosting provider records your IP address, the time of the request and user-agent information about your browser and device in Cloudflare Pages’ standard access logs.
For details, see the privacy policy in Korean.
Calculations run in your browser without sending your calculation inputs or results to the server. See “Privacy on this page” for information about access data.