Solve systems of equations with step-by-step elimination, substitution, and matrix methods. Supports 2x2 and 3x3 systems with solution checks.
Enter integers, decimals, or fractions (e.g. 1/2)
Last column is the constants vector (right-hand side).
0x + 0y = 0
0x + 0y = 0
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A system of equations calculator helps you solve multiple linear equations at once to find unknown variables like x, y, and z. This tool supports 2x2 and 3x3 systems, classifies whether the system has one solution, infinitely many solutions, or no solution, and shows every row operation step so you can learn and verify your work.
A system of equations is a set of equations that share the same variables and must all be true at the same time. In linear algebra, systems are commonly represented as an augmented matrix and solved using elimination, substitution, or row reduction. The final result can be unique, infinite, or inconsistent depending on rank and constraints.
Standard Form
Every row swap, scale, and elimination operation is displayed so students can follow the exact solving path and check classwork or exam preparation.
You get the outputs users care about most: solution type, variable values, free variables, and rank, not just a final matrix.
Switch between matrix, elimination, and substitution-oriented views while using one consistent and reliable calculation engine.
Inputs like 1/2 and -3/4 are supported, and exact arithmetic reduces rounding errors common in manual decimal-only workflows.
Check your manual elimination or substitution steps and confirm whether your final x, y, z values are correct before submitting assignments.
Model constrained scenarios such as pricing mixes, resource allocation, and balancing equations with multiple unknowns.
Solve small linear systems that appear in statics, circuits, and process balancing where multiple conditions must hold simultaneously.
Use real-world presets to understand when systems become inconsistent or underdetermined, and how free variables appear in solutions.
No solution means the equations conflict with each other. In matrix form, this appears as a contradiction row such as 0x + 0y = nonzero, which cannot be true.
Infinitely many solutions occur when at least one equation is dependent on others, leaving one or more free variables. The system is consistent but underdetermined.
For many 2x2 problems, substitution is intuitive. For larger systems, elimination and matrix row-reduction are typically faster and less error-prone.
RREF is the reduced row echelon form of the augmented matrix. It is a standardized final form that makes solution type and variable values easy to read directly.
Yes. Enter values like 1/2, -3/4, or 5/3. The calculator keeps exact fractional arithmetic during reduction for cleaner and more accurate results.