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System of Equations Solver

Type the coefficients of two equations in x and y, or three in x, y and z, to get the exact solution with Cramer's rule - every determinant shown - or a clear explanation when the system has no solution or infinitely many.

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Put each equation in the form ax + by (+ cz) = d. A missing variable has coefficient 0. Fractions (1/2) and decimals are fine.

Solution
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Steps (Cramer's rule)

    Worked example

    Solve 2x + 3y = 13 and x − y = −1.

    1. Determinant of the coefficientsD = 2 × (−1) − 3 × 1 = −5
    2. Replace the x column with 13 and −1Dx = 13 × (−1) − 3 × (−1) = −10, x = −10 ÷ −5 = 2
    3. Replace the y columnDy = 2 × (−1) − 13 × 1 = −15, y = −15 ÷ −5 = 3
    4. Check in both equations2(2) + 3(3) = 13 ✓ 2 − 3 = −1 ✓

    One solution, none, or infinitely many

    Each linear equation in two variables is a straight line, and the solution is where the lines cross.

    • The lines cross once - one solution. The determinant D is not 0.
    • The lines are parallel - no solution. D is 0 and the equations contradict each other, for example x + y = 2 and 2x + 2y = 5.
    • The lines are the same - infinitely many solutions. D is 0 and one equation is a multiple of the other, such as x + y = 2 and 2x + 2y = 4.

    With three variables each equation is a plane, and the same three cases apply. When D = 0 the solver compares the rank of the coefficients with the rank of the full system to tell "none" from "infinitely many".

    What this calculator does not do

    • Two or three equations with the same number of unknowns; larger systems are not supported.
    • Linear equations only - no x², xy or other non-linear terms.
    • When there are infinitely many solutions it says so but does not write the general solution in parameter form.

    Frequently asked questions

    How does Cramer's rule work?
    Compute D, the determinant of the coefficients. Then replace one variable's column with the right-hand sides and compute that determinant, Dx. The variable is Dx ÷ D. Repeat for each variable.
    What does it mean when D = 0?
    Then the system does not have exactly one solution. Either the equations contradict each other (parallel lines, no solution) or they describe the same line or plane (infinitely many solutions). The calculator checks which by comparing ranks.
    How do I enter an equation like 2x = y + 3?
    Rearrange it so the variables are on the left and the number on the right: 2x − y = 3. Then enter 2 and −1 as the coefficients and 3 as the right-hand side. A missing variable has coefficient 0.
    Can I use fractions and decimals?
    Yes. Enter 1/2 or 0.25. All arithmetic is done with exact fractions, so the answer has no rounding error.
    Is elimination or substitution better than Cramer's rule?
    They all give the same answer. Elimination is usually quicker by hand for three equations; Cramer's rule is the easiest to show as a short list of steps, which is why it is used here.

    Related free tools

    Quadratic Formula CalculatorExact roots of ax² + bx + c = 0 (radicals, complex), vertex and factored form.
    Slope CalculatorSlope, distance, midpoint, angle and the line's equation from two points.
    Fraction CalculatorAdd, subtract, multiply, divide and simplify fractions, with each step shown.
    GCF & LCM CalculatorGCF (HCF) and LCM of 2-10 numbers with prime factors and Euclid's steps.

    Step-by-step guide: How to solve systems of equations →

    Built and checked by Taimur Hassan Siddiqui, who has more than ten years in computer-science education. The formulas on this page are covered by an automated test suite, and the worked example on this page is one of its cases. Last updated 7 October 2026.