System of Equations Calculator
Solve a system of 2 or 3 linear equations with Cramer’s rule — enter each equation’s coefficients and constant to get x, y and z.
How it works
Cramer’s rule: xᵢ = det(Aᵢ) ÷ det(A), where Aᵢ is A with column i replaced by the constants
Cramer’s rule solves a square linear system using determinants. Compute the determinant of the coefficient matrix, then for each variable replace that variable’s column with the column of constants and take the new determinant; the variable equals the ratio of the two. If the coefficient determinant is zero, the equations are either inconsistent or dependent, so there is no single solution. It is elegant for 2 and 3 unknowns, which is exactly what this calculator handles.
Worked example
Take 2x + y = 5 and x − y = 1, entered as "2, 1, 5; 1, -1, 1". The coefficient determinant is 2×(−1) − 1×1 = −3. Replace the x-column with the constants: det[5, 1; 1, −1] = −6, so x = −6 ÷ −3 = 2. Replace the y-column: det[2, 5; 1, 1] = −3, so y = −3 ÷ −3 = 1. The solution is x = 2, y = 1.
Sources: Cramer’s rule — Wikipedia
Frequently asked questions
What if the determinant is zero?
A zero coefficient determinant means there is no unique solution — the lines (or planes) are parallel or coincide, giving either no solution or infinitely many. Cramer’s rule can’t single one out, so the calculator reports it.
How do I enter my equations?
Write each equation as its coefficients followed by the constant on the right side. For 2x + y = 5 enter "2, 1, 5"; for x − y = 1 enter "1, -1, 1". Separate the two equations with a semicolon.
What about equations not in this form?
Rearrange first so all variables are on the left and the constant is on the right, in a consistent x, y, z order. Missing variables get a coefficient of 0 to hold their place.
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This calculator is for educational purposes. Double-check important results before acting on them.