How to solve a system of equations
- Type one equation per line, like 2x + 3y = 7. Use single letters for the unknowns (x, y, z, w…). Terms can be on either side: y = 2x - 1 works.
- Read the solution as exact fractions, with decimals when they aren’t whole numbers.
- Check the steps below: the augmented matrix and the row operations that eliminate each unknown in turn.
Elimination in practice
Take x + y = 5 and x − y = 1. Subtracting the second equation from the first removes x: 2y = 4, so y = 2, and then x = 3. Gauss-Jordan elimination does the same thing in a fixed order that works for any number of equations: it clears the first unknown from every other equation, then the second, and so on, until each row holds one unknown.
The augmented matrix is just the coefficients written in a grid, with the right-hand sides in the last column. R₂ − 2R₁ means “replace row 2 with row 2 minus twice row 1”.
How many solutions to expect
With as many independent equations as unknowns there is exactly one solution. Fewer independent equations than unknowns give infinitely many, and contradictory equations give none. Three planes in space can meet in a point, along a line, or not at all; the solver tells you which.