How to use it
- Type one equation on each line in Equations. Use single letters for the unknowns, such as x and y.
- Read Solution. If it says there are infinitely many or no solutions, read the message as it stands.
- Check Reduced system and, for two or three equations in as many unknowns, Cramer's rule.
- Go through How it is worked out to see each elimination step.
Gaussian elimination
The equations are written as a matrix of coefficients with the constants at the end. Gaussian elimination, in the Gauss-Jordan form, swaps equations, multiplies one by a number, and subtracts a multiple of one equation from another, until each unknown with a pivot stands alone with a coefficient 1.
The calculator uses exact fractions, so no step is rounded. The solution is then read from the reduced system.
2x + y − z = 8
−3x − y + 2z = −11
−2x + y + 2z = −3 → x = 2, y = 3, z = −1Unique, infinite and no solution
A system has one solution when every unknown has a pivot. If an equation reduces to 0 = c with c not 0, there is no solution. If some unknowns have no pivot, they are free: they can take any value, and the others are written in terms of them.
For two or three equations in as many unknowns with a nonzero determinant, Cramer's rule gives each unknown as a ratio of two determinants: x = Dx ÷ D, y = Dy ÷ D and so on.
What the calculator does not cover
- Systems of more than six equations or more than six unknowns, and equations that are not linear, such as x² = 4.
- Unknowns written as x/2 or in brackets: write the number in front, as 0.5x or (1/2)x.
- Decimal approximations and numerical solvers for very large systems.
Worked example: 2x + y − z = 8, −3x − y + 2z = −11, −2x + y + 2z = −3
After elimination the reduced system reads x = 2, y = 3 and z = −1. Each unknown has its pivot, so the solution is unique.
Cramer's rule gives the same answer: D = −1, Dx = −2, Dy = −3 and Dz = 1, so x = (−2) ÷ (−1) = 2, y = (−3) ÷ (−1) = 3 and z = 1 ÷ (−1) = −1.
| Result | |
|---|---|
| x | 2 |
| y | 3 |
| z | −1 |
| Determinant D | −1 |
Frequently asked questions
How do I write an equation for the calculator?
Put one equation on each line, with the unknowns on the left and a number on the right, such as 2x + 3y = 7. Write a number in front of a letter for its coefficient, and write the sign of each term.
What does it mean when there are infinitely many solutions?
The equations depend on each other, so at least one unknown can take any value. The solution then gives the other unknowns in terms of it, such as x = 2 − y.
Why does the calculator say there is no solution?
The equations contradict each other. Elimination reaches a line that reads 0 = c with c not 0, which cannot be true, so no values of the unknowns solve every equation.
When is Cramer's rule shown?
It is shown for two or three equations in the same number of unknowns when the determinant is not 0. In other cases the solution comes from elimination alone, and Cramer's rule does not apply.
Can I use decimals in the equations?
Yes. Write them with a point, such as 0.5x + y = 1. The calculator reads them exactly, so 0.5 is 1/2 and the answer is not rounded.
Are the equations I type sent to a server?
No. The solution is worked out in your browser, and nothing you type is sent to us.
Sources
Figures and text last checked on Tuesday, November 10, 2026.