Math & Stats
Systems of Equations Solver
Solve 2×2 or 3×3 systems of linear equations. Get the solution, classification (consistent/inconsistent/dependent), and step-by-step working.
Enter coefficients and click Calculate
Solution Methods
Cramer's Rule (2×2):
a₁x + b₁y = c₁
a₂x + b₂y = c₂
D = a₁b₂ − a₂b₁
Dx = c₁b₂ − c₂b₁
Dy = a₁c₂ − a₂c₁
x = Dx/D, y = Dy/D
Classification:
D ≠ 0 → Consistent (unique solution)
D = 0, Dx=Dy=0 → Dependent (infinite solutions)
D = 0, otherwise → Inconsistent (no solution)FAQ
Frequently asked questions about systems of equations
What is a system of linear equations?
A system of linear equations is two or more equations with the same variables. The solution is the set of values that satisfies all equations simultaneously. For example, x + y = 5 and 2x − y = 1 has solution x=2, y=3.
What does it mean if a system is inconsistent?
An inconsistent system has no solution — the equations represent parallel lines (2D) or parallel planes (3D) that never intersect. For example, x + y = 3 and x + y = 5 has no solution.
What is a dependent system?
A dependent system has infinitely many solutions — the equations represent the same line or plane. For example, x + y = 3 and 2x + 2y = 6 are the same line, so every point on it is a solution.
What is Cramer's Rule?
Cramer's Rule solves systems using determinants. For a 2×2 system, x = det(Ax)/det(A), y = det(Ay)/det(A), where Ax replaces the x-column of the coefficient matrix with the constants column.
When should I use elimination vs substitution?
Use elimination when coefficients are easy to cancel by adding/subtracting equations. Use substitution when one variable is easily isolated (coefficient of 1 or −1). Both yield the same answer.
Can this solve 3-variable systems?
Yes, this calculator supports both 2-variable (2 equations) and 3-variable (3 equations) systems. For 3×3 systems, it uses Gaussian elimination and Cramer's Rule with step-by-step working.