Linear Equation System Solver

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system of two equations and two unknowns

System of 2 equations and 2 unknowns: solve online

A system of two equations and two unknowns is the most common case taught in schools, usually with the variables called x and y. This page opens the calculator with an example of that type already loaded, ready to be swapped for your own system.

Substitution, elimination, or matrix: three paths, the same answer

A 2x2 system is usually taught with two hand methods: substitution (isolate one variable in one equation and plug it into the other) and elimination (multiply one or both equations by a number so a variable cancels out when the two rows are added). For the example "2x + 3y = 12" and "x - y = 1": isolating x in the second equation, x = 1 + y, and substituting into the first, 2(1+y) + 3y = 12, gives 5y = 10, so y = 2 and x = 3. The Gaussian elimination this calculator uses is essentially the elimination method, just organized as a matrix and generalized to any number of equations and unknowns, without depending on which variable happens to be easiest to isolate.
For a square 2x2 system like this one, Cramer's rule also works directly: det(A) = (2)(-1) - (3)(1) = -5, det(A₁) (substituting the x column with the constant terms) = (12)(-1) - (3)(1) = -15, so x = -15 / -5 = 3, and similarly y = 2. All three approaches (substitution, elimination/Gauss, Cramer) always land on the same answer, because they describe the same mathematical fact in different ways.

A 2x2 system solved three different ways

By substitution

Input: x = 1 + y, 2(1+y) + 3y = 12
Expected output: y = 2, x = 3

Isolating x in the second equation and swapping it into the first.

Check against both original equations

Input: 2(3) + 3(2) = 12; 3 - 2 = 1
Expected output: 12 = 12; 1 = 1

Plugging x=3 and y=2 back in, both equations hold exactly.

Full tool FAQ

How should I type the equations?

One equation per line, with an equals sign separating the two sides, for example "2x + 3y = 8" or "x - y = -1". The calculator accepts an implicit coefficient ("x" means the same as "1x"), a negative sign, unknowns with any name, a variable appearing on both sides of the equals sign ("2x + 1 = x + 4"), and a variable that simply doesn't appear in one of the equations, it is treated as a zero coefficient automatically.

What's the difference between the Gaussian elimination and Cramer's rule shown here?

What does a "consistent and dependent" system mean?

How do I know a system is inconsistent, rather than that I typed something wrong?

Why does the answer show up as a fraction instead of a decimal number?

How many unknowns can a system have?

What does the "ill-conditioned" system warning mean?

Is my data saved on any server?

Frequently asked questions

Does this calculator solve by substitution or by matrix?

By matrix, using Gaussian elimination and Cramer's rule, both shown side by side. The result is always the same you'd reach by hand substitution, but the matrix method generalizes to systems with more than two unknowns without changing technique.

What if the two equations use different letters instead of x and y?

No problem, the calculator accepts any unknown name (a and b, m and n, whatever you use), it identifies the variables from what you typed, it does not expect fixed names.

Next steps

Substitution, elimination, or matrix: three paths, the same answer

A 2x2 system is usually taught with two hand methods: substitution (isolate one variable in one equation and plug it into the other) and elimination (multiply one or both equations by a number so a variable cancels out when the two rows are added). For the example "2x + 3y = 12" and "x - y = 1": isolating x in the second equation, x = 1 + y, and substituting into the first, 2(1+y) + 3y = 12, gives 5y = 10, so y = 2 and x = 3. The Gaussian elimination this calculator uses is essentially the elimination method, just organized as a matrix and generalized to any number of equations and unknowns, without depending on which variable happens to be easiest to isolate.

For a square 2x2 system like this one, Cramer's rule also works directly: det(A) = (2)(-1) - (3)(1) = -5, det(A₁) (substituting the x column with the constant terms) = (12)(-1) - (3)(1) = -15, so x = -15 / -5 = 3, and similarly y = 2. All three approaches (substitution, elimination/Gauss, Cramer) always land on the same answer, because they describe the same mathematical fact in different ways.

Full tool FAQ

How should I type the equations?

What's the difference between the Gaussian elimination and Cramer's rule shown here?

What does a "consistent and dependent" system mean?

How do I know a system is inconsistent, rather than that I typed something wrong?

Why does the answer show up as a fraction instead of a decimal number?

How many unknowns can a system have?

What does the "ill-conditioned" system warning mean?

Is my data saved on any server?

Frequently asked questions

Does this calculator solve by substitution or by matrix?

What if the two equations use different letters instead of x and y?

No problem, the calculator accepts any unknown name (a and b, m and n, whatever you use), it identifies the variables from what you typed, it does not expect fixed names.