Linear System Solver
Solve a system of two linear equations in two unknowns (a₁x + b₁y = c₁ and a₂x + b₂y = c₂) to find the x and y that satisfy both at once. Use it for circuit problems, mixture problems, or any two-variable algebra.
Last updated: September 2026
Formula below · 2 sources (mathworld.wolfram.com, Wikipedia) · Updated Sep 2026
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About this calculator
A system of linear equations consists of two or more equations sharing the same unknowns. For a 2×2 system a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the solution uses Cramer's Rule: x = (c₁b₂ − c₂b₁) / det and y = (a₁c₂ − a₂c₁) / det, where det = a₁b₂ − b₁a₂. If det = 0, the system either has no solution (inconsistent) or infinitely many solutions (dependent). Gaussian elimination achieves the same result by row-reducing the augmented matrix [A|b] to upper-triangular form and back-substituting. These methods underpin engineering, economics, and computer graphics wherever multiple constraints must be satisfied simultaneously.
How to use
Suppose you have: 2x + 3y = 12 and x − y = 1. Enter coefficients a₁ = 2, b₁ = 3, c₁ = 12, a₂ = 1, b₂ = −1, c₂ = 1. Compute det = (2)(−1) − (3)(1) = −2 − 3 = −5. Then x = (12·(−1) − 1·3) / (−5) = (−12 − 3) / (−5) = −15 / −5 = 3. And y = (2·1 − 1·12) / (−5) = (2 − 12) / (−5) = −10 / −5 = 2. Check: 2(3) + 3(2) = 12 ✓ and 3 − 2 = 1 ✓.
Frequently asked questions
What does it mean when the linear system has no unique solution?
When the determinant of the coefficient matrix equals zero, the equations are either parallel (no intersection, inconsistent system) or identical (infinite intersections, dependent system). In either case, Cramer's Rule and standard Gaussian elimination cannot produce a single unique answer. You can detect which case applies by checking whether the augmented matrix is also rank-deficient; if it is, infinitely many solutions exist along a line or plane.
How does Gaussian elimination work for solving linear systems?
Gaussian elimination transforms the augmented matrix [A|b] into row echelon form through a sequence of row operations: swapping rows, multiplying a row by a scalar, and adding a multiple of one row to another. Once the matrix is upper-triangular, back-substitution yields the values of each variable from the bottom row upward. This method is numerically stable with partial pivoting and scales to large systems where Cramer's Rule becomes computationally expensive.
When should I use Cramer's Rule versus matrix methods for linear systems?
Cramer's Rule is elegant and easy to apply by hand for 2×2 or 3×3 systems, giving each variable as a ratio of determinants. However, its computational cost grows factorially with system size, making it impractical beyond small systems. Matrix methods like Gaussian elimination or LU decomposition are preferred for larger systems because they scale as O(n³) and are the foundation of numerical solvers used in engineering software.