What is "determinant = 0" geometrically?

The two lines are PARALLEL — they have the same slope. Either they're identical (infinite solutions) or they never intersect (no solution).

Is Cramer's rule efficient?

For 2×2 and 3×3 it's fine. For n×n with n > 3, Gaussian elimination is O(n³) while Cramer's is O(n⁴) due to the n! determinant computation — Gaussian wins by orders of magnitude on large systems.

Cramer G (1750), "Introduction à l'analyse des lignes courbes algébriques". Modern presentation: Strang G (2016) "Introduction to Linear Algebra" 5th ed (Wellesley-Cambridge Press); Anton & Rorres, "Elementary Linear Algebra".

What is "determinant = 0" geometrically?

The two lines are PARALLEL — they have the same slope. Either they're identical (infinite solutions) or they never intersect (no solution).

Is Cramer's rule efficient?

For 2×2 and 3×3 it's fine. For n×n with n > 3, Gaussian elimination is O(n³) while Cramer's is O(n⁴) due to the n! determinant computation — Gaussian wins by orders of magnitude on large systems.

Cramer G (1750), "Introduction à l'analyse des lignes courbes algébriques". Modern presentation: Strang G (2016) "Introduction to Linear Algebra" 5th ed (Wellesley-Cambridge Press); Anton & Rorres, "Elementary Linear Algebra".

System of 2 Linear Equations Solver

Solve { a·x + b·y = e ; c·x + d·y = f } via Cramer's rule. Handles unique / no-solution / infinite-solutions cases.

Result

Unique solution

x = 1.000000, y = 2.000000

Determinant = -17.0000 ≠ 0 → exactly one solution.

Equation 12x + 3y = 8
Equation 25x + -1y = 3
Determinant D = ad − bc-17.000000
D_x = ed − bf-17.000000
D_y = af − ec-34.000000
x = D_x / D1.00000000
y = D_y / D2.00000000
Verify (eq 1)2·1.0000 + 3·2.0000 = 8.0000 (should = 8)
Verify (eq 2)5·1.0000 + -1·2.0000 = 3.0000 (should = 3)

Step-by-step

Cramer's rule for { ax+by=e ; cx+dy=f }: D = ad − bc, D_x = ed − bf, D_y = af − ec; x = D_x/D, y = D_y/D.
D = 2·-1 − 3·5 = -17.0000.
D_x = 8·-1 − 3·3 = -17.0000.
D_y = 2·3 − 8·5 = -34.0000.
x = -17.0000 / -17.0000 = 1.000000; y = -34.0000 / -17.0000 = 2.000000.

How to use this calculator

Write both equations in standard form: a·x + b·y = e.
Enter the six coefficients. The tool handles unique, no-solution, and infinite-solution cases automatically.
Read the "Verify" lines to confirm the answers satisfy both equations.

About this calculator

A 2×2 linear system is the simplest form of linear algebra: two equations, two unknowns. Cramer's rule (Gabriel Cramer, 1750) solves it via determinants: for {ax+by=e; cx+dy=f}, x = (ed−bf)/(ad−bc) and y = (af−ec)/(ad−bc), provided the denominator D = ad−bc ≠ 0. The three cases: D ≠ 0 → unique solution; D = 0 with D_x = D_y = 0 → infinite solutions (dependent system, lines coincide); D = 0 with D_x or D_y ≠ 0 → no solution (inconsistent system, parallel lines). Cramer's rule is elegant but for larger systems (n > 3) Gaussian elimination is much faster computationally — the rule is mainly a teaching aid past 2×2.