🔡System of Equations Calculator
Solve a system of 2 or 3 linear equations using Cramer's rule. Handles unique solutions, no solution (inconsistent), and infinitely many solutions (dependent). Shows the determinant, step-by-step Cramer's rule, and verification by substitution.
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Solution
x = 3, y = 2
Solution Values
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System of Equations Calculator: Solve 2×2 and 3×3 Linear Systems
A system of linear equations has a unique solution when the determinant D ≠ 0. For a 2×2 system a₁x + b₁y = c₁ and a₂x + b₂y = c₂, Cramer's rule gives: x = (c₁b₂ − c₂b₁) / (a₁b₂ − a₂b₁) and y = (a₁c₂ − a₂c₁) / (a₁b₂ − a₂b₁).
Key formula: D = a₁b₂ − a₂b₁ — if D = 0, the system has no solution or infinite solutions
| Condition | Geometric meaning | Solutions |
|---|---|---|
| D ≠ 0 | Lines intersect at one point | Unique solution |
| D = 0, D_x ≠ 0 or D_y ≠ 0 | Lines are parallel | No solution |
| D = 0, D_x = 0, D_y = 0 | Lines are identical | Infinite solutions |
This calculator uses Cramer's rule — a determinant-based formula that elegantly expresses the solution of Ax = b as ratios of determinants. It works for systems with a unique solution and automatically detects when no unique solution exists.
How Cramer's Rule Works
For a 2×2 system, D = a₁b₂ − a₂b₁ is the determinant of the coefficient matrix. To find x, replace the x-column with the constants and compute D_x = c₁b₂ − c₂b₁. Then x = D_x/D. For y, replace the y-column: D_y = a₁c₂ − a₂c₁, and y = D_y/D. For 3×3, the same principle applies with 3×3 determinants computed by cofactor expansion.
Geometric Interpretation
Each equation in a 2×2 system represents a line in the xy-plane. A unique solution is the intersection point of two non-parallel lines. A 3×3 system represents three planes in 3D space; a unique solution is the point where all three planes meet.
Frequently Asked Questions
How do you solve a system of 2 equations with 2 unknowns?
Three methods work for 2×2 systems: (1) Substitution — solve one equation for x in terms of y, substitute into the other. (2) Elimination — multiply equations to make coefficients match, then add/subtract to eliminate one variable. (3) Cramer's rule — compute D = a₁b₂ − a₂b₁; x = (c₁b₂ − c₂b₁)/D; y = (a₁c₂ − a₂c₁)/D. Example: 2x+3y=12, x−y=1. D = 2(−1)−1(3) = −5. D_x = 12(−1)−1(3) = −15. D_y = 2(1)−12(1) = −10. x = −15/−5 = 3, y = −10/−5 = 2.
What does it mean when a system of equations has no solution?
A system has no solution (inconsistent) when the equations represent parallel lines (2D) or parallel planes (3D) — they never intersect. Algebraically, this happens when D = 0 but the numerator determinants D_x or D_y are non-zero. Example: x + y = 3 and x + y = 5 — the same left-hand side but different right-hand sides, so no point satisfies both simultaneously. The lines are parallel (slope = −1 for both) but have different y-intercepts.
What does infinite solutions mean in a system of equations?
A system has infinitely many solutions (dependent) when the equations represent the same line (2D) or intersecting planes sharing a line (3D). All three determinants D, D_x, D_y equal zero. Example: x + y = 3 and 2x + 2y = 6 — the second equation is just twice the first, so they represent the same line. Any point on that line satisfies both equations: (0,3), (1,2), (2,1), etc. The solution set is typically expressed as {(t, 3−t) : t ∈ ℝ} using a parameter t.
How do you solve a 3×3 system of equations?
For 3×3 systems, the most common methods are: Gaussian elimination (row reduce the augmented matrix to row echelon form, then back-substitute) and Cramer's rule (compute 4 determinants: D, D_x, D_y, D_z; then x=D_x/D, y=D_y/D, z=D_z/D). Example: x+y+z=6, 2x−y+z=3, x+2y−z=2. D = det([[1,1,1],[2,−1,1],[1,2,−1]]) = 1(1−2)−1(−2−1)+1(4+1) = −1+3+5 = 7. D_x = det([[6,1,1],[3,−1,1],[2,2,−1]]) = 7, so x=1. Similarly y=2, z=3.