What does it mean if a linear system has no solution?

The equations contradict each other: geometrically, the planes they describe have no common point. Elimination reveals this when it produces an impossible statement such as 0 = 5. A system with infinitely many solutions instead produces 0 = 0, meaning one equation adds no new information.

Why does queue length explode near full utilization?

Random arrivals bunch up, and a server that is nearly always busy has little spare capacity to catch up after a burst. In the basic M/M/1 model, the average number in the system is ρ/(1 − ρ), which grows without limit as utilization ρ approaches 1.

When is the simplex method used in practice?

It is used for production planning, diet and blending problems, transportation and scheduling, wherever the goal and constraints are linear. Problems that require whole-number answers need integer programming, which builds on linear programming but is harder to solve.

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Advanced Equations calculators

Q: Is there a formula for equations of degree five or higher?

Not a general one in terms of radicals. The Abel-Ruffini theorem proves that no formula using only arithmetic and roots solves every quintic. Specific equations can still be solved, and numerical methods find roots of any degree to high accuracy.

This category holds solvers that go beyond a standard algebra course: quartic equations, three-variable linear systems, exponential and logarithmic equations, partial fraction decomposition, piecewise functions, the discrete Fourier transform, linear programming with the simplex method, and queuing theory. They are aimed at college math, engineering and operations research problems.

8 tools · Updated October 21, 2026

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xyz3-Variable System of Equations SolverEquations and functions2x+y−z=8, −3x−y+2z=−11, −2x+y+2z=−3x = 2, y = 3, z = −1

X[k]Discrete Fourier Transform (DFT) CalculatorOptimization, queues and signalsDFT of 1 1 1 1 −1 −1 −1 −1 at 1,000 HzPeak at 125 Hz

bˣExponential & Logarithmic Equation SolverEquations and functions2 · 3ˣ = 18x = 2

x⁴Partial Fraction Decomposition CalculatorEquations and functions(3x + 5)/(x² + 3x + 2)1/(x + 2) + 2/(x + 1)

x⁴Piecewise Function Calculator & EvaluatorEquations and functionsf(1): x² if x < 0, 2x + 1 if 0 ≤ xf(1) = 3, jump at x = 0

x⁴Quartic Equation SolverEquations and functionsx⁴ − 2x³ − 7x² + 8x + 12 = 0x = −2, −1, 2, 3

x⁴Queuing Theory CalculatorOptimization, queues and signals50 arrivals/h, 5 servers at 12/h each3.72 min wait in line

x⁴Simplex Method CalculatorOptimization, queues and signalsMax 3x₁+5x₂; x₁≤4, 2x₂≤12, 3x₁+2x₂≤18z = 36 at x₁ = 2, x₂ = 6

Which one do I need?

I need to solve three linear equations in x, y and z3-Variable System of Equations Solver I have a fourth-degree polynomial equation to solveQuartic Equation Solver The unknown is in an exponent or inside a logarithmExponential & Logarithmic Equation Solver I need to split a rational function into partial fractions before integratingPartial Fraction Decomposition Calculator I want to evaluate or graph a function defined in piecesPiecewise Function Calculator & Evaluator I want to maximize profit or minimize cost subject to linear constraintsSimplex Method Calculator I want average wait times and queue lengths for a service with arrivals and serversQueuing Theory Calculator I have sampled data and want its frequency spectrumDiscrete Fourier Transform (DFT) Calculator

Higher-degree equations and linear systems

Quartic equations are the highest degree with a general solution in radicals; beyond degree four, the Abel-Ruffini theorem shows no such general formula exists. The Quartic Equation Solver returns all four roots, real and complex, which is useful in mechanics and optics where quartics arise naturally.

A system of three linear equations has a unique solution, no solution or infinitely many, depending on whether the equations are independent and consistent. The 3-Variable System of Equations Solver uses elimination and reports which case applies, so you can tell a contradictory system from a dependent one.

Rewriting functions to make them workable

Partial fraction decomposition rewrites a rational function as a sum of simpler fractions. It is the standard step before integrating rational functions or taking inverse Laplace transforms, and the Partial Fraction Decomposition Calculator handles linear, repeated and irreducible quadratic factors.

Exponential and logarithmic equations are solved by taking logs of both sides or by rewriting a log equation in exponential form. Always check the domain afterward, since a logarithm of zero or a negative number is undefined. Piecewise functions, such as tax brackets or shipping rates, need each rule applied only on its own interval, which the Piecewise Function Calculator evaluates and graphs.

Optimization, queues and spectra

Linear programming finds the best value of a linear objective subject to linear constraints, such as maximizing profit from limited labor and materials. The optimum always lies at a corner of the feasible region, and the simplex method moves between corners until it cannot improve. The Simplex Method Calculator shows each tableau so you can follow the pivots.

Queuing theory predicts waiting times and queue lengths from the arrival rate and service rate. The key quantity is utilization: as it approaches 100%, waits grow very quickly, which is why a service that is busy most of the time can still have long lines. The Discrete Fourier Transform Calculator, meanwhile, breaks a sampled signal into the frequencies it contains.

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