Solve 2×2 and 3×3 linear systems by substitution, elimination, or Cramer's rule with full steps. Handles unique solutions, no solution (parallel lines), and infinite solutions (same line).
Lines intersect → Unique solution (x₀, y₀)
Unique Solution: The two lines cross at exactly one point. The determinant D ≠ 0. We get specific values for every variable.
No Solution (Parallel Lines): The lines have the same slope but different y-intercepts — they never meet. The determinant D = 0 and the equations are inconsistent. You'll see a contradiction like 0 = 5.
Infinite Solutions (Same Line): Both equations describe the exact same line — one is a multiple of the other. D = 0 and the equations are dependent. The solution is the entire line, expressed as a parametric set.
D = a₁b₂ − a₂b₁
x = Dₓ/D, y = Dᵧ/D
Dₓ = c₁b₂ − c₂b₁
Dᵧ = a₁c₂ − a₂c₁
For the system a₁x + b₁y = c₁ and a₂x + b₂y = c₂, Cramer's Rule uses determinants to find each variable directly.
D is the determinant of the coefficient matrix. If D = 0, the system has no unique solution.
Dₓ replaces the x-column with the constants. Dᵧ replaces the y-column with the constants.
One-on-one tutoring builds the intuition for choosing the right method — we work through your actual homework and tests so elimination, substitution, and Cramer's rule become second nature.