System of Equations Calculator
The system of equations calculator allows you to solve systems of two or three linear equations and tells you what your solution looks like, even if there are an infinite number of them.
Solutionx = 2, y = 1
x2
y1
Determinant-3non-zero, so the solution is unique
Check equation 15 = 5
Check equation 21 = 1
By elimination
| Determinant | 2×-1 − 1×1 = -3 |
| x (Cramer) | (5×-1 − 1×1) ÷ -3 = 2 |
| y (Cramer) | (2×1 − 1×5) ÷ -3 = 1 |
The method
a₁x + b₁y = c₁ and a₂x + b₂y = c₂
Three possible outcomes
Two straight lines either cross once, run parallel and never meet, or coincide entirely. The determinant a₁b₂ − a₂b₁ distinguishes them: non-zero means a unique solution, zero means either none or infinitely many depending on the constants.
Elimination, substitution and Cramer’s rule all reach the same answer; they differ only in bookkeeping.