System of Equations Solver
Solve 2, 3 or 4 linear equations by elimination with exact fractions and every row operation shown, or find out there is no solution or infinitely many.
The interactive tool loads here when JavaScript runs.
About this toolWhat is the System of Equations Solver?
Type your equations the way you would write them, such as 2x + 3y − z = 5, with terms on either side of the equals sign, fractions like x/2 and brackets like 2(x − y). The solver rearranges each one into standard form, builds the augmented matrix and solves it by elimination, showing every row operation and the matrix after each step.
All arithmetic is done with exact fractions, so an answer of 6/5 stays 6/5 instead of becoming 1.2000000001. The result is checked by substituting it back into every equation, and for square systems the determinant and Cramer's rule values are shown as a second, independent check.
Not every system has one answer. If the equations contradict each other the solver says there is no solution and shows the impossible row, such as 0 = 1. If they overlap, it identifies the free variables and writes the infinite family of solutions in terms of a parameter t.
- Equations typed in natural form, with terms on both sides
- Exact fraction arithmetic throughout
- Gauss-Jordan elimination with every row operation and matrix
- Detects no solution and infinitely many solutions with a parametric form
- Determinant, rank and Cramer's rule cross-check
- Substitution check of every equation
How to use the System of Equations Solver
- Type one equation per box. Use single letters (x, y, z, w) or indexed names (x1, x2) for the unknowns.
- Add or remove boxes for 2, 3 or 4 equations. You can also try the 2 × 2, 3 × 3 and 4 × 4 examples.
- Read the solution as exact fractions with decimals underneath, or the no-solution or infinitely-many message.
- Follow the elimination steps: each shows the row operations used and the augmented matrix that results.
- Check the substitution panel, where every equation is evaluated with the solution, and the Cramer's rule cross-check.
Worked examples
Two equations
Input
2x + 3y = 7, x − y = 1
Output
x = 2, y = 1
Swapping the rows puts a 1 in the pivot, then R2 − 2·R1 leaves 5y = 5. Back in the first row, x = 1 + y = 2.
Classic 3 × 3
Input
x + y + z = 6, 2y + 5z = −4, 2x + 5y − z = 27
Output
x = 5, y = 3, z = −2
The determinant is −21, and Cramer's rule gives x = −105 / −21 = 5, matching elimination.
No solution
Input
x + y = 2, 2x + 2y = 5
Output
Inconsistent
Subtracting twice the first equation leaves 0 = 1. The two lines are parallel.
Infinitely many
Input
x + y + z = 6, x − y + 2z = 5, 2x + 3z = 11
Output
x = 11/2 − (3/2)t, y = 1/2 + (1/2)t, z = t
The third equation is the sum of the first two, so only two are independent and z is free.
Background and insight
A system of linear equations asks for values that satisfy every equation at once. Geometrically, two equations in x and y are two lines: they cross once (one solution), are parallel (none) or are the same line (infinitely many). Three equations in three unknowns are planes in space with the same three possibilities.
Elimination, formalised as Gaussian elimination, adds multiples of one equation to another so that unknowns disappear one at a time. The method appears in the Chinese text The Nine Chapters on the Mathematical Art, around two thousand years ago, and is still the basis of numerical linear algebra libraries. This solver uses the Gauss-Jordan variant: each pivot is scaled to 1 and cleared from every other row, which leaves the reduced row echelon form where the answers can be read directly.
The rank of the matrix decides the outcome. If the coefficient matrix and the augmented matrix have the same rank and it equals the number of unknowns, there is exactly one solution. If the augmented matrix has a higher rank, a row has reduced to 0 = a non-zero number and there is no solution. If the ranks match but are smaller than the number of unknowns, the leftover unknowns are free and there are infinitely many solutions. For a square system, a non-zero determinant is the same as full rank.
Common use cases
Algebra homework
Check answers to elimination and substitution problems and compare your row operations with a full worked solution.
Linear algebra courses
See rank, reduced row echelon form, determinants and parametric solutions for small systems.
Word problems
Solve mixture, ticket-price, investment and coin problems once they are written as equations.
Engineering and science
Solve small circuit (Kirchhoff), force-balance and chemical-equation-balancing systems exactly.
Tips and limitations
- Every term must be linear: xy, x² or 1/x are rejected because elimination only works for linear equations.
- Write coefficients next to variables (3x) or with a multiplication sign (3*x); x/2 means one half of x.
- More equations than unknowns is fine; the solver checks whether the extra equations agree.
- For larger systems or matrix operations such as inverses, use the Matrix Calculator.
- An answer with a free parameter is still a complete answer: every value of t gives a valid solution.
Frequently asked questions
How do you solve a system of equations by elimination?
Multiply one equation and add or subtract it from another so that one variable cancels. Repeat until each equation has one variable left, then read off or back-substitute the values. The solver shows each of these row operations on the augmented matrix.
How can I tell if a system has no solution?
During elimination a row reduces to 0 = a non-zero number, such as 0 = 1. That means the equations contradict each other; for two variables, the lines are parallel.
What does infinitely many solutions mean?
Some equations are combinations of others, so there are fewer independent equations than unknowns. The leftover unknowns are free, and the solution is a family written with a parameter, like z = t, x = 11/2 − (3/2)t.
What is the augmented matrix?
It is the table of coefficients with the right-hand sides added as a last column. Elimination works on its rows, which is the same as working on the equations but with less writing.
What is the difference between Gaussian and Gauss-Jordan elimination?
Gaussian elimination clears entries below each pivot and then back-substitutes. Gauss-Jordan also clears entries above each pivot and scales pivots to 1, so the final matrix shows the answers directly. This solver uses Gauss-Jordan.
When should I use Cramer's rule?
For small square systems with a non-zero determinant, where each unknown equals a ratio of determinants. It is a neat check but needs far more arithmetic than elimination as the system grows, so it is shown here as a cross-check only.
Can I enter equations with fractions or decimals?
Yes. Coefficients like 0.25x, 3/4y or x/2 are converted to exact fractions, and answers stay exact, with decimals shown alongside.
Can there be more equations than unknowns?
Yes. The solver eliminates as usual; if the extra equations agree with the others you get the solution, and if any of them conflict you get no solution.
Is there an in-depth guide to the System of Equations Solver?
Yes. “Solving Systems of Linear Equations by Elimination, Step by Step” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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