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Free Online Simultaneous Equation Solver, Two to Six Unknowns

Solve a system of two to six simultaneous equations, typed the way you would write them rather than as a grid of coefficients. Systems with no solution and systems with infinitely many are both identified and explained, which is where most solvers give an error or a wrong answer instead.

Free Forever Nothing Uploaded Up to Six Unknowns All Three Outcomes

Free Online Simultaneous Equation Solver, Two to Six Unknowns

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🧩 Simultaneous Equation Solver Two to six unknowns · typed as written · all three outcomes named

The sum runs in your browser. Nothing you type is sent anywhere.

Equations read —

Unknowns found —

🧩 The equations

One per line, written as you would on paper. Any letters work, brackets are fine, and the multiplication sign can be left out.

Equations, one per line

Try one of these

The fourth and fifth are the two cases where other solvers give an error or a meaningless number.

The solution

—

What kind of answer—

Equations and unknowns—

Rank of the system—

Determinant—

Largest error on substitution—

How each equation was read, and what it comes to when the answers are substituted back in. Every row should balance.

#	As read	Left side	Right side

The system after elimination. A row of zeros with a number on the end means no solution; a row of all zeros means one equation told us nothing new.

Recent systems, kept in this browser only.

Nothing here yet Press Log to history on the Solve tab to keep a system here.

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How to Solve Simultaneous Equations

Three steps, and the third tells you which kind of answer you have.

Type one equation per line Written as you would on paper. Any letters can be the unknowns, brackets are fine, and you do not need to arrange them into any particular order.

Read the answer for each unknown Where the answers are not whole numbers the exact fractions are given, because two thirds is a better answer than 0.666667.

Look at what kind of answer it is One solution, none, or infinitely many. The last two are real outcomes with real meanings, and the row that tells you which is the first thing to check if the answer surprises you.

What to Know About Systems of Equations

Three outcomes, not one, and how to tell which you have.

No solution means the equations contradict each other. x plus y equals one, and twice x plus twice y equals three. The second says the same combination is both one and one and a half. Geometrically the lines are parallel and never meet. This is a correct and complete answer, and it is often the useful one: it tells you your constraints cannot all be satisfied at once.

Infinitely many means one equation told you nothing new. x plus y equals one, and twice x plus twice y equals two: the second is the first doubled. There are not enough independent equations to pin down each unknown, so there is a whole family of solutions. The page names which unknowns are free to vary and gives one particular solution, which is more useful than a bare statement that there are many.

A solver that divides by zero here gives you a number that means nothing. Both the above cases make the determinant zero, and the classic formula divides by it. Some tools return infinity, some return a very large number that looks like a real answer, and some return an error. The honest response is to recognise the case and name it, which needs the rank rather than just the determinant.

The pivot is chosen rather than taken in order. Elimination that always uses the first row loses accuracy badly when that row has a tiny coefficient in it. Taking the largest available coefficient at each step keeps the arithmetic stable. On a system with coefficients a hundred million apart, in-order elimination can get the answer completely wrong while looking perfectly healthy.

Whole number answers are shown as whole numbers. Elimination on integer coefficients leaves results like 2.9999999999999996, and printing that is misleading: it makes a correct answer look like an approximation. Values within the rounding tolerance are snapped to the whole number, and nothing outside the tolerance is touched.

More equations than unknowns is allowed, and often informative. Four equations in three unknowns usually has no solution, because the fourth is unlikely to agree with the other three. That is worth knowing rather than being refused. Fewer equations than unknowns always gives infinitely many, and the free unknowns are listed.

Key Features & Capabilities

What it does, and what it deliberately does not.

Typed as written One equation per line with brackets and any letters, instead of filling a grid of coefficients.

All three outcomes One solution, none, or infinitely many, each named, with the free unknowns listed in the last case.

Substitution check Every equation evaluated at the answer, with the largest discrepancy reported so you can see it is genuinely zero.

Up to six unknowns Elimination with pivoting rather than a two by two formula, so larger systems work and stay accurate.

The eliminated system shown The reduced form, where a row of zeros against a number is exactly what no solution looks like.

Nothing is uploaded It runs in this tab and keeps working offline.

About Simultaneous Equations

Simultaneous equations are several conditions that have to hold at the same time, and solving them means finding values that satisfy all of them together. Two equations in two unknowns is the version everybody meets at school, and it has a tidy formula. Beyond that the formula becomes unwieldy and the practical method is elimination: use one equation to remove an unknown from the others, and repeat.

The thing worth understanding is that there are three possible outcomes rather than one. Usually the equations pin down a single answer. Sometimes they contradict each other, in which case no values can satisfy them all and the honest answer is that there is no solution. And sometimes one equation is just another one in disguise, in which case there are not enough independent conditions and a whole family of answers works.

Those last two cases are where tools fail, and they fail in an unhelpful way. Both make the determinant zero, and the standard formula divides by the determinant. Depending on how the arithmetic rounds you get infinity, an error, or a very large number that looks like a genuine answer. Distinguishing properly needs the rank of the system, which is the number of genuinely independent equations in it, so that is what is computed here and reported alongside the determinant.

Two smaller decisions matter for the answers being right. The elimination picks the largest available coefficient at each step rather than working in the order you typed, because a small leading coefficient destroys accuracy and the damage is invisible. And where the answers are whole numbers they are printed as whole numbers; elimination on integers naturally produces things like 2.9999999999999996, and showing that makes a correct answer look doubtful. Only values within the rounding tolerance are adjusted, and the substitution check shows what the real residual is.

Frequently Asked Questions

Unknowns, outcomes, and what to do when there is no single answer.

Up to six unknowns and up to six equations. The two by two case has a tidy formula, but beyond that elimination is the practical method and that is what runs here, with pivoting so it stays accurate.

That the equations contradict each other, so no set of values satisfies them all. With two unknowns it means the lines are parallel. It is a complete answer, and in applied work it usually means your constraints are incompatible.

That at least one equation duplicates information already in the others, so there are not enough independent conditions. The page lists which unknowns are free to take any value and gives one particular solution to work from.

No. Any single letters work and the answers are reported with the letters you used. The page tells you which letters it found, which is also a quick way to spot a typo that invented an extra unknown.

Yes. Usually that has no solution, because the extra equations are unlikely to agree with the rest, and knowing that is useful. If they do all agree, you get the single answer as normal.

Because a determinant of zero is the signal that the system does not have a single answer. It does not tell you which of the other two cases you are in, which is why the rank is shown next to it.

No. The parsing and the elimination both happen in your browser and the page works offline once loaded.