How to use this calculator
Three linear equations may have one solution, no solution, or infinitely many solutions. Row operations simplify the system without changing its solution set. This calculator uses scaled Gaussian elimination with partial pivoting, shows the reduced matrix, and checks residuals when a unique solution exists.
The method
Enter the values in the labeled fields, check the units or selected mode, and choose Calculate. The result includes the applicable working and a breakdown or visual when useful. Change an input and calculate again to update the result.
Equation 1: a, b, c, d: 2, 1, -1, 8 Equation 2: a, b, c, d: -3, -1, 2, -11 Equation 3: a, b, c, d: -2, 1, 2, -3
Linear system solution: x = 2, y = 3, z = -1
A unique intersection satisfies all three equations.
Scope and limitations
Three equations in three unknowns only. Nearly dependent systems are numerically sensitive. An infinite-solution result identifies the classification but does not output a complete symbolic parameterization.
Common questions
How do I enter a missing variable?
Use a zero coefficient. For x + z = 4, enter 1, 0, 1, 4.
What does no solution mean?
At least one equation contradicts the others. In row reduction, this appears as zero on the left equaling a nonzero right side.
References & further reading
External references provide background, not an endorsement of this implementation. See our methodology for numerical conventions and limitations.