System of Equations Solver
Solve 2×2 and 3×3 systems of linear equations using Cramer's rule and elimination with step-by-step solutions.
System of equations
Updates as you typeSystem size
How many unknowns? ?
Scenario
(pre-filled examples)
Paste equations ?
Equations
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Solution walkthrough
Pick a methodAugmented matrix, step by step
Each line is one equation. Their intersection is the solution.
Formula
Cramer's rule (2×2)
D = a1b2 − b1a2
x = (c1b2 − b1c2) / D
y = (a1c2 − c1a2) / D
D = a1b2 − b1a2
x = (c1b2 − b1c2) / D
y = (a1c2 − c1a2) / D
Cramer's rule (3×3)
Compute D, D_x, D_y, D_z by replacing the corresponding coefficient column with the right-hand side.
x = Dx/D, y = Dy/D, z = Dz/D.
Compute D, D_x, D_y, D_z by replacing the corresponding coefficient column with the right-hand side.
x = Dx/D, y = Dy/D, z = Dz/D.
- ai, bi, ci
- Coefficients of the equations (left-hand side).
- D
- Determinant of the main coefficient matrix.
- Dx, Dy, Dz
- Determinants formed by replacing the x, y or z column with the right-hand side.
- Elimination
- Adding/subtracting equations to remove one variable at a time.
- RREF
- Reduced row-echelon form — the final matrix shape that reads off the solution.
Worked example — your numbers
- D = —
- Dx = —
- Dy = —
- Dz = —
- Solution = —
Cramer's rule computes solutions directly from determinants; elimination (Gaussian) is more practical for larger systems. When D = 0 the system either has no solution (inconsistent) or infinitely many (dependent).