Simultaneous Equations Solver
Equation 1: a₁x + b₁y = c₁
Equation 2: a₂x + b₂y = c₂
Method
Uses Cramer's rule to solve the system of linear equations.
Solve two linear equations in x and y, written as a₁x + b₁y = c₁ and a₂x + b₂y = c₂, with Cramer's rule. When there is no single answer the solver says why: parallel lines never meet, and two equations for the same line share every point.
Formula
Worked Example
The determinant D = 2·(−1) − 1·1 = −3 is not zero, so there is exactly one solution: x = (11·(−1) − 1·1) ÷ −3 = 4 and y = (2·1 − 1·11) ÷ −3 = 3. Check: 2·4 + 3 = 11 and 4 − 3 = 1.
Frequently Asked Questions
What does it mean when the determinant is zero?
The two lines have the same slope. If one equation is a multiple of the other (x + 2y = 3 and 2x + 4y = 6) they are the same line and there are infinitely many solutions; otherwise (x + 2y = 3 and 2x + 4y = 7) they are parallel and there is none.
Which method is best: elimination, substitution or Cramer's rule?
All three give the same answer. Elimination is usually quickest by hand, substitution suits a system where one variable is already isolated, such as y = 3x − 2, and Cramer's rule gives a direct formula that also shows at once whether a unique solution exists.
Can it solve three equations with three unknowns?
No, it handles two equations in two unknowns. The same Cramer's rule extends to 3 × 3 systems with 3 × 3 determinants, though Gaussian elimination is less work by hand.