System Of Equations Calculator

What this calculator does

A system of equations is a set of equations that all have to be true at the same time, sharing the same unknowns. Solving it means finding the values that satisfy every equation at once. This solves systems of 2 or 3 variables.

Enter your equations and it returns the solution. It runs right here in the browser.

Using the calculator

1. Choose 2 or 3 equations, for two or three unknowns.
2. Enter each equation's coefficients and its constant term.
3. Press Calculate.

It returns the value of each unknown. Reset clears everything.

What a system of equations is

On its own, a single equation like 2x + 3y = 13 has endless answers, since many pairs of x and y fit it. Pair it with a second equation, though, and you are asking for the values that work for both at once, which usually pins things down to a single answer.

Geometrically it is easy to picture. Each equation in two variables is a line, and the solution is the point where the lines cross. With three variables, each equation is a flat plane in space, and the solution is the point where all three planes meet. Solving the system is finding that meeting point.

How it solves them

This tool uses Gaussian elimination, the systematic version of the elimination method. It combines the equations to remove one variable at a time, whittling the system down to a simple triangular shape where the last equation has just one unknown. From there it works backwards, substituting each solved value into the equation above, until every unknown is found. It also picks the largest available coefficient to work with at each step, a trick called partial pivoting that keeps the arithmetic stable, and it leans on the math.js library for reliable computation. It all runs in your browser.

One answer, none, or infinitely many

Not every system has a single tidy answer, and it is worth knowing the three possibilities. Most systems have exactly one solution, the lines or planes crossing at a single point. But sometimes the equations describe parallel lines that never meet, and then there is no solution, a system that contradicts itself. And sometimes two equations are secretly the same line, so they overlap everywhere and there are infinitely many solutions. A good solver recognises all three, rather than forcing a single answer that is not really there.

A worked example

Solve 2x + 3y = 13 and x − y = -1.

1. The second equation gives x = y − 1.
2. Put that into the first: 2(y − 1) + 3y = 13, which is 5y − 2 = 13, so 5y = 15 and y = 3.
3. Back-substitute: x = 3 − 1 = 2.

So x = 2, y = 3. Checking both: 2(2) + 3(3) = 4 + 9 = 13, and 2 − 3 = -1. Both hold, so the two lines cross at the point (2, 3).

The different ways to solve a system

There is more than one route to the answer, and each has its own tool. The elimination method adds or subtracts equations to cancel a variable, and is the technique most people learn first. Cramer's rule solves it with determinants. Reduced row echelon form reduces the matrix of the system directly, and the matrix inverse solves it in one multiplication. They all reach the same solution; which is handiest depends on the size of the system and what you already have.

Questions people ask

What is a system of equations?

A set of equations sharing the same unknowns that must all be true together. The solution is the set of values satisfying every equation at once.

How does the calculator solve it?

By Gaussian elimination: removing one variable at a time to reach a triangular system, then substituting back to find each unknown.

Can a system have no solution or many?

Yes. Parallel lines give no solution, identical lines give infinitely many, and crossing lines give exactly one.

How many variables can it handle?

Two or three, matching the number of equations you choose.

What does the solution mean geometrically?

For two variables, the point where the lines cross. For three, the point where the three planes meet.

References

A note on where this comes from. Solving a system of linear equations by systematically eliminating variables is Gaussian elimination, named after Carl Friedrich Gauss, though the method appears far earlier, in the ancient Chinese text the Nine Chapters on the Mathematical Art. For further reading, see System of linear equations.

1. Gaussian elimination, the method of reducing a system to triangular form and solving by back-substitution.
2. math.js, a JavaScript mathematics library, used here for reliable computation of the solution.