How the system of equations calculator works
The calculator reads each equation with the same parser used by the graphing and calculus tools, moves everything to one side, and extracts the coefficient of each variable. Those coefficients become the rows of an augmented matrix [A | b], which is then reduced to reduced row echelon form with exact fractions. The shape of the reduced matrix decides the outcome: a pivot in every variable column means one solution, a missing pivot means a free variable and infinitely many solutions, and a contradictory row means none. Every row operation is listed, so the page doubles as a step-by-step Gaussian elimination guide.
Worked example: the default three equations
The defaults are 2x + y - z = 8, -3x - y + 2z = -11, and -2x + y + 2z = -3. The augmented matrix has rows (2, 1, -1 | 8), (-3, -1, 2 | -11), (-2, 1, 2 | -3). Scaling row 1 by 1/2 and adding multiples of it to the other rows clears the x column. The y column's pivot is then 1/2 in row 2; scaling it and clearing the column leaves row 3 with a pivot for z. Clearing the z column above it finishes the job, and the final matrix reads x = 2, y = 3, z = -1. Substituting back: 2(2) + 3 - (-1) = 8, -3(2) - 3 + 2(-1) = -11, and -2(2) + 3 + 2(-1) = -3, so all three equations check.
The three possible outcomes
| Reduced matrix shows | Outcome | Example |
|---|---|---|
| A pivot in every variable column | Exactly one solution | 2x + 3y = 7, x - y = 1 gives x = 2, y = 1 |
| Some variable column has no pivot | Infinitely many solutions | x + y + z = 3, 2x + 2y + 2z = 6 gives x = 3 - s - t, y = s, z = t |
| A row 0 = c with c not 0 | No solution | x + 2y = 4, 2x + 4y = 9 |
Reading a parametric solution
When a system has free variables, the calculator names them s, t, u, and w in order and writes each unknown in terms of them. The constant part is one particular solution (all parameters set to 0), and the parameter parts are directions along which the solution set extends: a line for one parameter, a plane for two. To get a specific solution, pick any numbers for the parameters. This matches the vector form x = p + s v1 + t v2 used in linear algebra courses.
Other ways to solve small systems
| Method | Best for | Drawback |
|---|---|---|
| Substitution | Two equations, one easily isolated variable | Messy with three or more unknowns |
| Elimination | Any size by hand | Easy to lose a sign; this page shows every step |
| Cramer's rule | Square systems with a nonzero determinant | Says nothing when the determinant is 0 |
| Matrix inverse | Many right-hand sides with the same A | Only for square, invertible A |
Entering equations
Variables are sorted alphabetically to form the columns of the matrix, so it does not matter which order the equations or terms come in. You may write a coefficient as a fraction (x/3 or (1/3)x) or a decimal, and implicit multiplication such as 2x or 3(x + y) works. A variable that appears in some equations but not others is treated as having coefficient 0 where it is missing. Use the selector to switch between two, three, and four equations; the number of unknowns is discovered from what you type.