System of Equations Calculator with Steps
Solve 2x2 or 3x3 linear systems using exact Gaussian-Jordan elimination. Enter coefficients as integers, decimals, fractions or scientific notation, then review the solution, determinant, ranks, row operations and checks.
Last Updated: July 26, 2026
Online Linear System Solver
Choose the system size. For each equation, enter the coefficient of every variable and the constant on the right side. The calculator forms the augmented matrix and reduces it exactly.
Your system solution will appear here.
Gaussian-Jordan elimination steps
- Enter a 2x2 or 3x3 coefficient system.
Reduced row echelon form
| x | y | Constant |
|---|---|---|
| Solve a system to see its reduced matrix. | ||
Equation checks
| Equation | Substituted left side | Right side | Status |
|---|---|---|---|
| A unique solution will be checked in every original equation. | |||
Recent calculations
- Your last six results will appear here.
How to Use This System of Equations Calculator
- Choose 2 equations and 2 variables, or 3 equations and 3 variables.
- Write each equation in coefficient form. For a 2x2 system, use ax + by = d. For a 3x3 system, use ax + by + cz = d.
- Enter each coefficient and right-side constant in its field. Enter 0 when a variable is absent from an equation.
- Select the decimal precision. Exact fractions are not changed by this setting.
- Select Solve System. Review the classification, exact solution, RREF matrix, row operations, ranks and equation checks.
Each field accepts a signed integer, finite decimal, fraction or scientific-notation value. Valid examples include 4, -2.5, 1/3 and 6e-2. Do not enter x, y, z, commas or a complete equation inside a coefficient field. The labels already assign every number to the correct variable.
What This Simultaneous Equations Calculator Solves
A system of linear equations asks for values that satisfy every equation at the same time. In two variables, a unique solution is an ordered pair (x, y). In three variables, it is an ordered triple (x, y, z). The calculator handles square 2x2 and 3x3 systems with exact rational coefficients.
The engine classifies three possible outcomes. A consistent independent system has one unique solution. A consistent dependent system has infinitely many solutions because at least one variable remains free. An inconsistent system has no solution because row reduction produces a contradiction.
This page is limited to linear systems. It does not solve x squared terms, products such as xy, trigonometric systems, inequalities, or systems with more than three variables. Use the single Equation Calculator for one equation in one variable. Matrix operations, graphing and nonlinear systems remain separate specialist tasks.
Coefficient Form and the Augmented Matrix
A 2x2 linear system can be written as a1x + b1y = d1 and a2x + b2y = d2. The six entered numbers form a coefficient matrix A and a constant column b. Combining them gives the augmented matrix [A | b].
a2x + b2y = d2
[A | b] = [[a1, b1 | d1], [a2, b2 | d2]]
A 3x3 system adds a z coefficient and a third equation. Keeping variables in the same column is essential. If an equation is x + z = 5, enter 1 for x, 0 for y, 1 for z and 5 for the constant.
Gaussian-Jordan Elimination
The calculator uses exact row operations to transform the augmented matrix into reduced row echelon form, usually called RREF. Gaussian elimination first creates pivots and zeros below them. Gauss-Jordan elimination continues until each pivot is 1 and every other value in a pivot column is 0.
Three elementary row operations preserve the solution set:
- Swap two rows.
- Multiply or divide one row by a nonzero number.
- Add a multiple of one row to another row.
The displayed steps identify each swap, pivot normalization and elimination. Because calculations use reduced fractions rather than binary floating-point values, a fraction such as 1/3 stays exact throughout the row reduction.
Unique, Infinite or No Solution
| Outcome | Rank test | RREF signal |
|---|---|---|
| Unique solution | rank(A) = rank([A | b]) = number of variables | Every variable column has a pivot. |
| Infinitely many solutions | rank(A) = rank([A | b]) but the common rank is smaller than the variable count | At least one variable has no pivot and becomes free. |
| No solution | rank(A) is smaller than rank([A | b]) | A row becomes 0 = a nonzero constant. |
For an infinite system, the calculator writes a parametric answer. For example, x + y = 2 may become x = 2 - t and y = t, where t is any real number. This represents every point on the shared line, not one selected point.
Worked 2x2 Example
Consider x + y = 3 and 2x - y = 0. The augmented matrix begins as [[1, 1 | 3], [2, -1 | 0]]. Eliminate the x entry in the second row by replacing row 2 with row 2 minus 2 times row 1.
[[1, 1 | 3], [0, -3 | -6]]
R2 = R2 / -3
[[1, 1 | 3], [0, 1 | 2]]
R1 = R1 - R2
[[1, 0 | 1], [0, 1 | 2]]
The unique solution is x = 1 and y = 2. Substitution verifies 1 + 2 = 3 and 2(1) - 2 = 0. The coefficient determinant is -3, so the coefficient matrix is invertible.
Worked 3x3 Example
Now use x + y = 3, 2x - y = 0 and x + y + z = 6. The first two equations still produce x = 1 and y = 2. Substituting those values into the third equation gives 1 + 2 + z = 6, so z = 3.
The RREF matrix is the 3x3 identity matrix followed by the constant column [1, 2, 3]. Every variable column has a pivot. Both ranks are 3, the determinant is -3 and all three original equations verify exactly.
Dependent and Inconsistent Examples
The system x + y = 2 and 2x + 2y = 4 is dependent because the second equation is twice the first. Row reduction produces one nonzero row and one zero row. Both ranks equal 1, which is smaller than the two-variable count, so one variable is free and infinitely many solutions exist.
Change the second constant from 4 to 5. The coefficient rows still match, but the constant does not follow the same multiple. Elimination produces 0x + 0y = 1. That false equation raises the augmented rank above the coefficient rank, so the system has no solution.
Determinant and Rank
For a square coefficient matrix, a nonzero determinant guarantees a unique solution. A zero determinant means the matrix is singular, so a unique solution is impossible. It does not tell you whether the remaining outcome is no solution or infinitely many solutions.
The rank comparison completes the classification. Rank counts pivot positions after row reduction. If adding the constant column creates an extra pivot, the system is inconsistent. If the two ranks match but do not reach the number of variables, free variables remain.
Exact Fractions and Decimal Rounding
Finite decimals are converted to exact fractions based on their written place value. The value 0.125 becomes 1/8, while 1/3 stays 1/3. Row operations reduce numerator and denominator pairs after every calculation.
The decimal selector affects only the separate approximation. A repeating result such as 1/3 is marked as approximate when rounded. Use the exact fraction for later algebra and round only when your assignment, measurement or application requires a stated number of decimal places.
How to Check a System Solution
For a unique solution, substitute the exact value of every variable into every original equation. Each left side must equal its own right-side constant. Checking only one equation is incomplete because a point may satisfy one line or plane without satisfying the complete system.
The calculator keeps an unchanged copy of the original coefficient matrix for this check. It multiplies each exact solution value by its original coefficient, adds the terms and compares the reduced fraction with the original constant. The equation-check table shows the substitution separately for every row.
An infinite solution needs a symbolic check. The calculator substitutes the constant and parameter coefficients from the displayed family and confirms that each parameter term cancels in every equation. For a no-solution result, verification takes a different form: an exact RREF row such as 0 = 1 proves that the system is inconsistent.
Input Rules and Calculator Limits
Every active coefficient field must contain one numeric value. Spaces around a fraction slash are accepted. Denominators cannot be zero. Each written number may contain up to 50 significant digits, and scientific exponents must stay from -100 through 100.
The tool solves only 2x2 and 3x3 square linear systems. Large exact intermediate values are limited to prevent an unresponsive page. Unsupported text receives a field-specific error, and the prior result is cleared when an input, system size or decimal setting changes.
Common System-Solving Mistakes
- Putting coefficients in different variable orders across the equations.
- Leaving a missing variable blank instead of entering a zero coefficient.
- Changing one entry during a row operation instead of applying the operation to the complete row.
- Dividing by a pivot that has become zero instead of swapping rows.
- Assuming determinant zero always means no solution.
- Stopping at a free variable without writing the full parametric solution.
- Rounding fractions during elimination and carrying error into later steps.
- Checking a result in only one equation instead of every original equation.
Method and Review Basis
The system classifications and two-variable methods follow OpenStax College Algebra 2e. The three-variable outcomes follow OpenStax three-variable systems. The row-reduction method follows OpenStax Gaussian elimination.
The calculator parses only the numeric coefficient fields and never executes them as program code. Exact calculations and recent history run in the current browser page. Your coefficients are not needed by a calculation server, and the recent-history list disappears when the page reloads.
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Frequently Asked Questions
What systems does this calculator solve?
It solves square linear systems with two equations and two variables or three equations and three variables.
Can I enter fractions and decimals?
Yes. Coefficients may be integers, finite decimals, fractions or scientific notation, and rational arithmetic stays exact.
What does a unique solution mean?
Every variable column has a pivot, so exactly one ordered pair or ordered triple satisfies all equations.
What does no solution mean?
Row reduction produces a contradiction such as 0 = 1, so no point satisfies every equation at the same time.
What does infinitely many solutions mean?
The system is consistent but has one or more free variables, so a parameter describes an entire family of solutions.
What is Gaussian-Jordan elimination?
It uses valid row operations to transform an augmented matrix into reduced row echelon form, where pivots and free variables are clear.
Why does the calculator show two ranks?
Comparing the coefficient rank with the augmented rank distinguishes a consistent system from an inconsistent one.
Does determinant zero mean no solution?
No. A zero determinant rules out a unique solution, but rank determines whether the system has no solution or infinitely many.
What should I enter for a missing variable?
Enter 0 for its coefficient so every equation keeps x, y and z in the correct columns.
Does the calculator verify the answer?
Yes. For a unique solution, it substitutes every exact variable value into every original equation and compares both sides.
Can it solve nonlinear systems?
No. Squares, variable products, radicals, trigonometric functions and other nonlinear terms are outside this coefficient-based solver.
Are my coefficients sent to a server?
No input is needed by a server to calculate the answer. Parsing, row reduction and recent history run in the current browser page.
Disclaimer: This calculator provides general mathematical results for learning and checking work. Verify coefficient order, model assumptions, notation and required rounding before using an answer for academic, scientific, financial, engineering or other important decisions.