Solve System Equation Calculator
System of Linear Equations Solver works out Solution type and Solution from Input style, Equations (one per line) and Variables (optional). The default figures on this page give solution of x = 6, y = 4; change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
7 steps
- System: x + y = 10 ; x - y = 2
- 2 equation(s) in 2 unknown(s): x, y.
- Eliminate y: multiply equation 1 by -1 and equation 2 by -1, then add.
- That leaves -2x = -12, so x = 6.
- Back-substitute into equation 1 to get y = 4.
- Solution: x = 6, y = 4.
- Check: substituting back gives residuals 0, 0 (all zero).
Solution by variabletable
| variable | value |
|---|---|
| x | 6 |
| y | 4 |
Cramer's rule numeratorstable
Reduced row echelon formtable
| 0 | 1 | 2 |
|---|---|---|
| 1 | 0 | 6 |
| 0 | 1 | 4 |
Its 5 worked examples are re-run on every release, last on 15 September 2026. What changed
Sources and references
2 sources
What the System of Linear Equations Solver calculates
Solves systems of linear equations of any size by Gauss-Jordan elimination, reporting the unique solution, the parametric family when the system is dependent, or that it is inconsistent — with substitution, elimination or Cramer’s rule working.
How to use this System of Linear Equations Solver
Enter Input style, Equations (one per line), Variables (optional) and Augmented matrix (row by row). Solution type updates as you type, and “Use a worked example” loads checked figures you can adjust. Change one figure at a time to see what drives the result.
How the result is worked out
Open “How this is worked out” beside the result for the step-by-step calculation behind Solution type, Solution and Solution by variable.
The method follows OpenStax, College Algebra 2e, Sections 9.1–9.8: Systems of Equations and Inequalities (substitution, elimination, Gaussian elimination, Cramer’s rule) and Golub & Van Loan, Matrix Computations, 4th ed., Chapter 3: General Linear Systems.
x + y = 10, x - y = 2 example
With Equations (one per line): x + y = 10 x - y = 2, the calculator returns x: 6, y: 4, Determinant of the coefficient matrix: -2. This example is checked against an independent source in the test suite.
What you can compare
Change one input at a time to see how it affects Solution type, Solution and Solution by variable. The inputs you can vary are Input style, Equations (one per line), Variables (optional), Augmented matrix (row by row), Number of unknowns and Method to show.
Results are general information, not financial, tax, legal, medical or engineering advice. Check any decision that matters with a qualified professional.
See something wrong? Report a problem with this calculator
Frequently asked questions
How to use a calculator to solve a system of equations?
Solves systems of linear equations of any size by Gauss-Jordan elimination, reporting the unique solution, the parametric family when the system is dependent, or that it is inconsistent — with substitution. Open “How this is worked out” beside the result to see each step for your own figures.
Can a TI-84 solve a system of equations?
Solves systems of linear equations of any size by Gauss-Jordan elimination, reporting the unique solution, the parametric family when the system is dependent, or that it is inconsistent — with substitution. Put your own figures in above and the working updates beside the result. For scale, the default figures on this page give x = 6, y = 4.
What's the easiest way to solve systems of equations?
Solves systems of linear equations of any size by Gauss-Jordan elimination, reporting the unique solution, the parametric family when the system is dependent, or that it is inconsistent — with substitution. On this page it is computed from Input style, Equations (one per line) and Variables (optional) and shown as Solution type, Solution and Solution by variable.
How to solve 3 system of equations calculator?
It needs Input style, Equations (one per line), Variables (optional), Augmented matrix (row by row), Number of unknowns and Method to show. Nothing else is required, and the figures stay in your browser.
What information do I need for the System of Linear Equations Solver?
You need Input style, Equations (one per line), Variables (optional), Augmented matrix (row by row), Number of unknowns and Method to show. The page opens with default figures that give solution of x = 6, y = 4, so you can see the scale before entering your own.
What does this System of Linear Equations Solver show?
It shows Solution type, Solution, Solution by variable, x, y, z, w and Determinant of the coefficient matrix, with the working and any warnings for your entries.