Systems of Equations Calculator
Solve a linear 2×2 or 3×3 system with exact reduced fractions, row operations, and a check against every original equation.
Private calculation: coefficients, solutions, steps, and export text stay on your device unless you choose Copy or Download TXT. ToolNeko does not put them in URLs, storage, cookies, requests, or analytics. Analytics receives only fixed tool and action identifiers.
Enter coefficients instead of a typed equation
Each card represents one equation. Put the number multiplying x, y, and—when 3×3 is selected—z in the labeled fields, then enter the value after the equals sign under right-hand side (RHS). For x + y = 7, enter 1, 1, and 7. Enter a zero explicitly when a variable is absent.
Follow exact elimination
The solver performs deterministic Gauss–Jordan elimination from left to right. It swaps with the first available lower row when a pivot is zero, reduces every fraction, and displays the augmented matrix after each operation. Decimal input is converted to its exact base-10 fraction before elimination; binary floating-point values do not decide the result.
Read the solution classification
A pivot for every variable gives one unique solution. In that case, the calculator substitutes the exact values into every original equation and shows both sides equal. A contradictory row such as 0 = 1 means no solution. Missing pivots without a contradiction mean infinitely many solutions, so the result uses free parameters t and u, plus v only when all three variables are unconstrained.
Input and calculation limits
Every field is required and accepts a signed integer from -9999 to 9999, a finite decimal with up to four places in the same range, or a fraction whose numerator is from -9999 to 9999 and denominator is from 1 to 9999. A field is limited to 32 pasted characters. Scientific notation, Unicode minus signs, mixed numbers, expressions, infinity, and zero denominators are rejected. A 512-bit intermediate safety limit stops unusually complex fraction growth without exposing a partial answer.
Focused linear-system scope
This page solves only square 2×2 and 3×3 linear systems entered as coefficients. It does not parse equation text, graph lines or planes, solve nonlinear or polynomial systems, handle inequalities, accept 4×4 matrices, call an AI service, save history, or upload values. Copy and Download TXT use the same current classification, exact answer or family, elimination steps, and checks shown on screen.
Frequently asked questions
How does the calculator tell whether a system has no solution?
Exact row reduction compares the coefficient and augmented ranks. If a row reduces to zero coefficients with a nonzero right side, such as 0 = 1, the equations contradict one another and the system has no solution.
What does infinitely many solutions mean?
It means at least one variable is free after elimination. The calculator names free parameters t, u, and—only for a fully unconstrained 3×3 system—v, then writes every pivot variable in terms of them instead of showing one incomplete ordered tuple.
Can I enter fractions and decimals?
Yes. Use integers from -9999 to 9999, decimals with up to four places, or fractions such as -3/4 with numerator and denominator within the displayed limits. The calculation keeps exact reduced fractions.
Can this solve three equations with three variables?
Yes. Choose 3 equations / 3 variables to enter the coefficients for x, y, and z, plus the right-hand-side value. The same exact elimination, classification, checks, copy, and TXT export flow is used.
Can I paste equations such as 2x + y = 5?
No. This focused version accepts coefficients in labeled fields rather than algebra expressions. It does not parse nonlinear equations, inequalities, polynomials, or systems larger than 3×3.
Are my equations uploaded or saved?
No. Solving, checking, copying, and TXT export happen in your browser. Coefficients and derived values are not put in URLs, storage, cookies, requests, or analytics.