System of equations calculator guide
A solution to a linear system must satisfy every equation simultaneously. Two equations in two variables can represent intersecting, parallel, or coincident lines; three equations in three variables similarly represent planes with one, none, or infinitely many common points.
The calculator uses Gaussian elimination with pivot-row selection on the augmented matrix. Comparing the coefficient rank with the augmented rank distinguishes an inconsistent contradiction from a dependent system with free variables.
How to solve a linear system
- Choose the size: Select two equations in two variables or three equations in three variables.
- Enter the augmented rows: Place variable coefficients in order and the equation constant in the separated last column.
- Solve: Select Solve system to perform partial-pivoting row reduction.
- Read the classification: A result may be unique, inconsistent with no solution, or dependent with infinitely many solutions.
Formula and variables
Elementary row operations preserve the solution set. Pivots determine rank and, when every variable column has a pivot, the unique solution.
A x = b; row-reduce [A | b] to reduced row-echelon form- A — Coefficient matrix
- Square 2 by 2 or 3 by 3 array of variable coefficients.
- x — Variable vector
- Unknown values x and y, or x, y, and z.
- b — Constant vector
- Right-side constants for the equations.
- rank — Independent-row count
- Number of pivot relationships after elimination.
Solve a 2x2 system
Solve x + y = 5 and 2x - y = 1.
- Equation 1 row
- 1, 1 | 5
- Equation 2 row
- 2, -1 | 1
- Add the equations after aligning coefficients to eliminate y
- 3x = 6, so x = 2
- Substitute into x + y = 5 to get y = 3
Result: The unique solution is x = 2, y = 3.
The ordered pair satisfies both original equations at the same time.
Understanding your results
Unique solution
The coefficient matrix has full rank, so every variable column has a pivot and one ordered pair or triple satisfies the system.
Infinite or inconsistent
Rank deficiency with no contradiction gives free variables and infinitely many solutions. A pivot in the constant column creates a contradiction and no solution.
Assumptions
- The system is linear and square with either two or three equations.
- Every coefficient and constant is a finite real number.
- Floating-point rank decisions use a scale-aware numerical tolerance.
Limitations
- The interface does not display parametric formulas for dependent systems.
- It does not solve nonlinear systems or systems larger than 3 by 3.
- Ill-conditioned systems may require higher-precision numerical software.
- Units and variable meanings must be interpreted outside the coefficient matrix.
Common mistakes
- Entering constants among the variable coefficients.
- Changing the variable order between rows.
- Assuming a zero determinant always means no solution rather than checking for dependence.
- Rounding coefficients before elimination.
- Checking a candidate solution in only one equation.
Practical use cases
Simultaneous constraints
Find values that satisfy two or three linear relationships at once.
System classification
Distinguish an intersection from parallel/inconsistent or coincident/dependent relationships.
Planning and decision guide
Preserve row meaning
Choose one variable order, use it for every row, and keep the right-side constants in the last column. Verify a unique result by substitution into all original equations.
Frequently asked questions
What is a unique solution?
It is one ordered pair or triple that satisfies every equation in the system.
What makes a system inconsistent?
Elimination produces a contradiction, meaning the augmented rank exceeds the coefficient rank and no common solution exists.
What makes a system dependent?
At least one equation does not add an independent constraint, leaving one or more free variables and infinitely many solutions.
Why use row swapping?
Partial pivoting moves a stronger nonzero coefficient into the pivot position and avoids division by a zero pivot when possible.
Can a 3x3 system have infinitely many solutions?
Yes. Dependent planes can share a line or be redundant, leaving free variables after elimination.
Sources and review
- Systems of Linear Equations: Two Variables — OpenStax College Algebra 2e. Accessed 2026-08-29.
- Systems of Linear Equations: Three Variables — OpenStax College Algebra 2e. Accessed 2026-08-29.
Reviewed 2026-08-29.