How to Use the System Solver
Equations
Use Equations mode to enter the coefficients and constants row by row. This mode is for structured coefficient input, not free-form text parsing.
A and b
Use A and b mode when you already have the coefficient matrix A and the constants vector b for Ax = b. The calculator keeps the two inputs separate and builds the augmented system internally.
Augmented Matrix
Use Augmented Matrix mode when your system is already written as [A|b]. The final column is treated as the constants column, and every earlier column is treated as a variable coefficient column.
How to Read the Result
Unique solution
A unique solution appears when the system is consistent and every variable column has a pivot. The Solution Summary lists one value for each variable.
No solution
No solution is reported when RREF contains a contradictory row such as [0 0 | 1]. That row means the equations cannot all be true at the same time.
Infinitely many solutions
Infinitely many solutions appear when the system is consistent but at least one variable is free. The Solution Summary uses parameters such as t1 or t2 to describe all solutions.
Square systems, underdetermined systems, and overdetermined systems can all be checked by RREF when the entered dimensions are valid.
Exact Values
The inputs support fractions such as 1/2, decimals such as 0.5, negative values, and constants such as pi and e. Empty cells follow the calculator's current behavior and are treated as 0.
Learn the Method
Want to understand every step?
Learn how equations become Ax = b, how to build an augmented matrix, and how RREF reveals the solution.
Related Tools
- RREF Calculator - row-reduce any matrix and review row-operation steps.
- Augmented Matrix Calculator - build or check an augmented matrix layout.
- Matrix Worksheet Generator - create printable practice sets with answer keys.
Frequently Asked Questions
Can I enter equations directly?
Yes. Use Equations mode to enter the coefficients and constants row by row. The mode uses structured coefficient fields rather than free-form text parsing.
Can I enter fractions and decimals?
Yes. You can enter fractions such as 1/2, decimals such as 0.5, negative values, and constants such as pi and e.
Can the calculator solve non-square systems?
Yes. The calculator can work with fewer equations than variables, more equations than variables, or the same number of each when the dimensions are valid.
How does it show no solution or infinitely many solutions?
The calculator row-reduces the augmented matrix to RREF. A contradictory row indicates no solution, while a consistent system with free variables indicates infinitely many solutions.
Can I enter A and b separately?
Yes. Use A and b mode when you already have the coefficient matrix A and the constants vector b. The calculator builds the augmented system internally.