🔗 Linear System Solver

Linear System Solver Ax = b

A system of linear equations Ax = b can be solved by Gaussian elimination, and this calculator determines whether the system has a unique, infinite, or no solution. Detects unique, infinite, or no solution. Shows augmented matrix RREF and back-substitution with full steps.

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System of Linear Equations Ax = b

Enter coefficient matrix A and constants b

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Coefficient Matrix A
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Constants b
Quick Examples

Solving Systems of Linear Equations

A system of linear equations is a set of equations sharing the same variables, like 2x + y = 5 and x - y = 1. Writing this as Ax = b lets us solve many equations at once using matrix methods, which is far faster than solving them one at a time by substitution.

$$Ax = b \quad \Rightarrow \quad x = A^{-1}b$$

Types of Solutions

  • Unique solution: rank(A) = rank([A|b]) = n. One exact solution vector x.
  • Infinite solutions: rank(A) = rank([A|b]) < n. Free variables exist.
  • No solution: rank(A) < rank([A|b]). System is inconsistent.

Solution Method

This solver uses augmented matrix RREF (Gauss-Jordan elimination), which systematically reduces [A|b] to [I|x] for unique solutions or identifies pivot/free variables for other cases.

Worked Example

For the system 2x + y = 5 and x - y = 1: written in matrix form, A = [[2,1],[1,-1]] and b = [5,1]. Solving gives x = 2 and y = 1 — you can check this works by substituting back into both original equations.

Applications

  • Circuit analysis in electrical engineering
  • Balancing chemical equations
  • Finding intersection points in economics and optimization problems

Frequently Asked Questions

This happens when the equations contradict each other, like x + y = 2 and x + y = 5. The calculator will identify this case and let you know no solution exists.
This happens when two or more equations describe the same relationship. The calculator will show you the general form describing every possible solution.
Not necessarily — this calculator can handle systems with more equations than unknowns or fewer, though a unique solution usually requires exactly as many independent equations as unknowns.

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