Algebra
System of Equations Solver
Solve simultaneous linear equations by entering the augmented matrix and applying Gauss–Jordan elimination.
Enter an augmented matrix (coefficients | constants) to solve Ax = b.
Gauss–Jordan reduction
Row operations transform the augmented matrix [A | b] into reduced row-echelon form, revealing the solution vector x.
How to use
- Enter each equation as a row of coefficients with the constant term last.
- Ensure every row has the same number of entries.
- Review the solved variables instantly or adjust the system if it’s singular.
Example
Input: 2 1 5\n1 −1 1
Output: x₁ = 2, x₂ = 1
Student-friendly breakdown
This walkthrough emphasizes the most searched ideas for System of Equations Solver: polynomial equation solver, polynomial root calculator, system of equations calculator, simultaneous equations solver. Start with the formula above, then follow the guided steps to double-check your work. For quick revision, highlight the givens, plug into the equation, and finish by verifying your units.
Need more support? Use the links below to open the long-form guide, browse additional examples, or hop into adjacent calculators within the same topic — each one is a quick way to double-check your work or handle a related question without starting from scratch.
Deep dive & study plan
System of Equations Solver: Solves linear systems using an augmented matrix. It's built around linear system, gaussian elimination, augmented matrix, so you can go from a raw question to a checked answer without switching tools.
The math behind it: Row operations transform the augmented matrix [A | b] into reduced row-echelon form, revealing the solution vector x.
To use it well: (1) Enter each equation as a row of coefficients with the constant term last. (2) Ensure every row has the same number of entries. (3) Review the solved variables instantly or adjust the system if it’s singular. Keep your units consistent as you go, and re-run a case you already know the answer to — it's the fastest way to catch a typo before it throws off a result you're relying on.
Worked example: entering 2 1 5\n1 −1 1 returns x₁ = 2, x₂ = 1. Try swapping in your own numbers next, especially a case you're unsure about, before you use this for something that matters.
Quick retention checklist
- Speak the formula aloud (or annotate it) so the relationships stick.
- Write each step in your own words and compare with the numbered list above.
- Swap in new numbers for the Example to make sure the calculator (and your logic) handles edge cases.
- Check at least one related calculator below — it's the fastest way to confirm your numbers still line up from a different angle.
FAQ & notes
Can it detect infinite solutions?
If the system lacks a unique solution the solver flags the input so you can inspect the equations manually.
How many variables can it solve?
Use up to six variables comfortably. Larger matrices may amplify floating-point error.
What formula does the System of Equations Solver use?
Row operations transform the augmented matrix [A | b] into reduced row-echelon form, revealing the solution vector x.
How do I use the System of Equations Solver?
Enter each equation as a row of coefficients with the constant term last. Ensure every row has the same number of entries. Review the solved variables instantly or adjust the system if it’s singular.