How to use the system of equations solver
- Choose 2 x 2 system or 3 x 3 system.
- Enter the coefficients of x, y (and z) and the constant on the right side of each equation, as in ax + by + cz = d. Use 0 for a missing variable.
- Read the result: a unique solution, no solution, or infinitely many solutions written with a parameter.
- Follow the row reduction steps to see each swap and subtraction, and check the answer by substituting it back.
Formula
Write the system as an augmented matrix [A | b]
Use row operations (swap, scale a row, subtract a multiple of another row) to reach reduced row echelon form
Rank of A = number of pivot columns
Unique solution if rank = number of unknowns, none if a row reads 0 = nonzero, infinitely many otherwise
Coefficients are read as exact decimals and all arithmetic uses exact fractions, so there is no rounding error and answers such as -1/5 appear as fractions with their decimal beside them. The determinant of the coefficient matrix is shown for a unique solution.
Worked example
2x + 3y = 13 and x - y = -1 have the unique solution x = 2, y = 3, and the determinant of the coefficients is -5.
The 3 x 3 system x + y + z = 6, 2y + 5z = -4, 2x + 5y - z = 27 has the unique solution x = 5, y = 3, z = -2.
x + 2y = 3 and 3x + y = 1 give exact fractions x = -1/5 and y = 8/5. The parallel lines x + y = 2 and x + y = 3 have no solution, and x + y = 2 with 2x + 2y = 4 describes the same line, so there are infinitely many solutions: x = 2 - t, y = t.
Three kinds of answers
A linear system can behave in only three ways. Two lines in a plane cross at one point, run parallel and never meet, or lie on top of each other. Three planes in space behave the same way in a more complicated manner. The solver reports which case applies and, for infinitely many solutions, describes them with one parameter t (a line of solutions) or two parameters t and s (a plane of solutions).
Why exact fractions
Floating-point arithmetic can turn a clean answer into something like 0.30000000000000004. This tool converts the numbers you type into exact fractions, so a result such as 8/5 is exact and the decimal 1.6 is shown alongside it. That is helpful for checking homework and for spotting singular systems, which depend on exact zeros.
Checking your answer
Substitute the values into each original equation. For x = 2 and y = 3 in 2x + 3y = 13, 4 + 9 = 13, which works. For a quadratic with one unknown instead, use the quadratic equation solver, and for fractions in your own working, the fraction calculator.
Assumptions and limits
- Only linear systems with as many equations as unknowns (2 x 2 or 3 x 3) are supported.
- Coefficients must be plain decimals up to 1e9 in size. Exponent notation is not accepted.
- Variables are named x, y and z. Parameters for infinite solution sets are named t and s.
- Steps list each row operation in the order the solver used. Another valid order would reach the same answer.
- Everything runs in your browser. Nothing you enter is sent anywhere.
Frequently asked questions
How do I solve a system of two equations?
Eliminate one variable by combining the equations, solve for the other, and substitute back. The solver does this with row operations and lists each step, so for 2x + 3y = 13 and x - y = -1 you get x = 2 and y = 3.
What does it mean when a system has no solution?
The equations contradict each other. Geometrically, two lines are parallel or three planes have no shared point. Row reduction shows a row that says 0 equals a nonzero number.
What does infinitely many solutions mean?
At least one equation is redundant, so the equations describe the same line or plane. Every point on it works. The solver writes the answer with a parameter, such as x = 2 - t and y = t.
What is the determinant for?
For a square system, a nonzero determinant means a unique solution exists. A zero determinant means the system is singular and has either no solution or infinitely many.
Can the solver handle decimals?
Yes. Decimals are converted to exact fractions before solving, so 0.1x + 0.2y = 0.3 is handled exactly with no floating-point error.