Enter a 2×2 or 3×3 linear system and solve it with Cramer’s rule. Get the unique solution — or learn the system has no solution or infinitely many — with the determinant and a graph.
Determinant
-2
Type
Unique solution
Calculator
Solution
x = 3, y = 2
The lines/planes meet at a single point.
Determinant
-2
Type
Unique solution
Cramer’s rule
D = -2
x = Dx / D = -6 / -2 = 3
y = Dy / D = -4 / -2 = 2
Graph
How the system is solved
Each equation is written as a₁x + b₁y (+ c₁z) = d. Cramer’s rule uses the determinant D of the coefficient matrix: each variable equals the determinant of the matrix with that variable’s column replaced by the constants, divided by D. A non-zero D guarantees exactly one solution.
When D = 0 the system is degenerate. If the equations are consistent they describe the same line or plane and there are infinitely many solutions; if they contradict each other there is no solution. The calculator decides between these by comparing the rank of the coefficient matrix with the rank of the augmented matrix.
What does the determinant tell me?
A non-zero determinant means the system has exactly one solution. A zero determinant means the lines or planes are parallel or identical, so there are either no solutions or infinitely many.
Can it solve 3×3 systems?
Yes. Switch to 3 variables to enter a 3×3 system; it is solved by Cramer’s rule with 3×3 determinants. The 2-variable mode also draws the two lines so you can see the intersection.
Why are there sometimes infinitely many solutions?
If one equation is a multiple of another (or a combination of the others), it adds no new information. The equations then describe the same line or plane, and every point on it is a solution.
Results are estimates. Verify with a professional for important decisions.
About this calculator
This calculator solves a 2x2 or 3x3 system of linear equations and tells you whether it has one unique solution, no solution, or infinitely many. Use it to check homework, verify hand-worked systems, or explore how changing a coefficient shifts the intersection of lines or planes.
How to read your results
The result card shows either the exact solution (x, y, and optionally z) or the system's classification — unique, none, or infinite. Below the card, the step-by-step breakdown displays the coefficient-matrix determinant and how each variable is recovered via Cramer's rule. For 2x2 systems a coordinate-plane graph plots both lines so you can see their intersection at a glance.
Worked example
Enter the 2x2 system: equation 1 is x + y = 5 and equation 2 is x - y = 1 (coefficients 1, 1, 5 and 1, -1, 1).
The determinant of the coefficient matrix is -2. Cramer's rule gives x = -6 / -2 = 3 and y = -4 / -2 = 2, so the unique solution is x = 3, y = 2. The graph shows the two lines crossing at (3, 2).
Frequently asked questions
What does it mean when the determinant is zero?
A zero determinant means the equations are not independent. The calculator then checks the augmented matrix: if its rank matches the coefficient matrix the lines (or planes) coincide and there are infinitely many solutions; if the ranks differ the system is inconsistent and has no solution.
What is Cramer's rule and when does it apply?
Cramer's rule expresses each variable as a ratio of determinants — the numerator replaces the variable's column in the coefficient matrix with the constant terms, and the denominator is the coefficient-matrix determinant. It applies only when the determinant is non-zero, i.e. the system has exactly one solution.
Can I solve a system with decimal or fractional coefficients?
Yes. Every coefficient cell accepts any finite decimal number. The solver works in floating-point arithmetic with a small tolerance for near-zero pivots, so results are accurate for typical homework problems and well-conditioned engineering systems.
How it's calculated
The solver extracts the n x n coefficient matrix A and the constant vector b from the input rows. It computes det(A) by cofactor expansion (Cramer's rule numerators are found by replacing each column of A with b). The system's classification is determined by comparing the rank of A with the rank of the augmented matrix [A|b] via Gaussian elimination with partial pivoting: rank(A) = rank([A|b]) = n means a unique solution, rank(A) = rank([A|b]) < n means infinite solutions, and rank(A) < rank([A|b]) means no solution. Sources: Wolfram MathWorld — Linear System of Equations; Khan Academy — Solving systems by elimination.
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