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System of Equations Solver

Solve 2 or 3 linear equation systems by substitution, elimination, Cramer or matrix. Detailed steps and graph.

① x + y =
② x + y =
Examples: 2x+y=5 / x−y=1 3x+2y=12 / x−y=1 x+y=4 / 2x−y=2 ∞ solutions ∅ sans solution x=3, y=5
① x + y + z =
② x + y + z =
③ x + y + z =
Examples: Exemple 1 → (2,3,-1) x=3 y=5 z=7 Exemple 2
↵ Enter to solve
Enter the coefficients and click Solve.

How to solve a system of equations?

A system of linear equations is a set of equations sharing the same unknowns. The solution is the set of values that simultaneously satisfy all equations. Depending on the case, a system may have a unique solution, infinitely many solutions, or no solution.

Substitution method

Isolate one unknown in one equation, then substitute into the others. Simple but can become complex with fractions. Ideal for 2×2 systems with simple coefficients.

Gaussian elimination and Cramer's rule

Gaussian elimination combines equations to progressively eliminate unknowns. Cramer's rule uses determinants: x = Det(Aₓ)/Det(A), y = Det(Aᵧ)/Det(A). These methods are systematic and apply to 2×2 and 3×3 systems.

Frequently asked questions

Cramer's rule is an algebraic method that uses determinants to solve linear equation systems. For a 2×2 system, x = Det(Aₓ)/Det(A) and y = Det(Aᵧ)/Det(A), where... Cramer's rule is an algebraic method that uses determinants to solve linear equation systems. For a 2×2 system, x = Det(Aₓ)/Det(A) and y = Det(Aᵧ)/Det(A), where A is the coefficient matrix, Aₓ replaces the first column with constant terms and Aᵧ replaces the second column. The method only works if Det(A) ≠ 0 (unique solution system).

A 2×2 system has no solution when the two lines are parallel but distinct: same slope but different y-intercepts. Mathematically, Det(A) = 0 but determinants Dx... A 2×2 system has no solution when the two lines are parallel but distinct: same slope but different y-intercepts. Mathematically, Det(A) = 0 but determinants Dx or Dy are non-zero. Geometrically, the lines never intersect. A system has infinitely many solutions when the equations are proportional (coincident lines): Det(A) = 0 and Dx = Dy = 0.

Gaussian elimination (pivot method) combines equations to progressively eliminate unknowns. Multiply one equation by a scalar and subtract it from another to ze... Gaussian elimination (pivot method) combines equations to progressively eliminate unknowns. Multiply one equation by a scalar and subtract it from another to zero out a coefficient. For a 3×3 system, first eliminate x from equations ② and ③ using ①, then eliminate y from ③ using ②. This gives z directly, then back-substitute to find y and x.

Yes, our solver accepts integer, decimal (e.g. 1.5, 0.25) and negative coefficients. Calculations are performed in floating point with 9 decimal places of preci... Yes, our solver accepts integer, decimal (e.g. 1.5, 0.25) and negative coefficients. Calculations are performed in floating point with 9 decimal places of precision. When the solution is a simple fraction (e.g. 1/3, 2/5), it is displayed in fractional form for clarity. The final verification confirms the solution satisfies all equations.

The graph represents each equation as a line in the plane. The system's solution is the intersection point of the two lines, shown in orange. If the lines are p... The graph represents each equation as a line in the plane. The system's solution is the intersection point of the two lines, shown in orange. If the lines are parallel (no intersection), the system has no solution. If the lines coincide (same line), the system has infinitely many solutions. The graph is automatically centered on the intersection point and adjusts its scale.

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