System of Two Linear Equations
Solve a1*x + b1*y = c1; a2*x + b2*y = c2 via Cramer's rule.
Example
You enter
- a1 (coefficient of x in row 1) 2
- b1 (coefficient of y in row 1) 3
- c1 (constant in row 1) 8
- a2 (coefficient of x in row 2) 1
- b2 (coefficient of y in row 2) -1
- c2 (constant in row 2) 1
You get
- Determinant -5
- x 2.2
- y 1.2
Checked against published worked examples from: Cramer (1750), standard linear algebra.
Details, formula, and sources
ScopeReports unique / infinite / no-solution; shows the determinant.
Formula or tabledet = a1*b2 - a2*b1. If det != 0: x = (c1*b2 - c2*b1)/det, y = (a1*c2 - a2*c1)/det (Cramer's rule). If det = 0 and (a1*c2 = a2*c1) and (b1*c2 = b2*c1): infinitely many solutions (same line). If det = 0 otherwise: no solution (parallel lines).
Source and editionStandard linear algebra; Cramer's rule per Gabriel Cramer, 'Introduction a l'analyse des lignes courbes algebriques,' 1750.
Free accessUniversal reference; covered in any algebra II / pre-calculus textbook.
What governsEstimate only. Readability formulas and similar metrics are derived from a representative population and have known edge-case noise. The classroom teacher governs final text selection, grade placement, and assessment decisions.
API field namesa1, b1, c1, a2, b2, c2, determinant, x, y
- Field real coefficients; complex coefficients are not supportedscope
- Consistency check cross-multiplication form avoids division by zero (a1*c2 == a2*c1 and b1*c2 == b2*c1)standard equivalence test