What is a system of equations calculator?
This system of equations calculator solves two linear equations in x and y, or three linear equations in x, y and z. A solution must satisfy every equation at the same time. Instead of returning unrelated answers for individual equations, the calculator works on the whole system and identifies whether there is one solution, no solution or infinitely many solutions.
Simultaneous equations is another common name for this task. The mathematics does not depend on a country, currency or unit system, so the same tool can support learners in the United States, United Kingdom, Canada and Australia. If your problem has just one unknown in one equation, the existing Linear Equation Calculator is a simpler choice. This page is not a general nonlinear equation solver.
How to use the simultaneous equations calculator
- Choose two or three variables to match your system.
- Rearrange each equation so all variable terms are on the left and its constant is on the right.
- Enter the signed coefficient of each variable and the right-side constant. Enter 0 when a variable is absent.
- Select Calculate, then inspect the classification, exact answers and working.
For 2x + y = 15, enter 2 for x, 1 for y and 15 on the right. For 3x − y = 5, the y coefficient is −1, not 1. Ordinary decimal entries are converted to exact fractions. A fraction must use a slash, such as 2/3; a denominator of zero is rejected. The Fraction Calculator can help with separate fraction arithmetic before you enter a coefficient.
All visible fields must be completed. The tool starts blank, and changing any input or switching system size clears the old answer. Values in the hidden third-variable fields are not used in two-variable mode. Reset clears both the entries and the result; Copy, Download and Print include the complete calculation explanation.
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For a separate arithmetic check, you may browse scientific calculators (paid link). Check the model’s supported functions; basic scientific devices do not necessarily solve systems or factor symbolic expressions. No purchase is required.
Elimination method and exact row operations
The solver represents the coefficients as an augmented matrix: one row per equation, with the right side in the final column. It uses Gauss–Jordan elimination to swap rows, divide a pivot row by its nonzero pivot and subtract multiples of that row from the other rows. Each performed operation is shown with the actual matrix that follows it.
These operations preserve the system's solutions. Exact rational arithmetic avoids treating a tiny nonzero coefficient as zero through a rounding tolerance. The working lists successful operations, not artificial filler steps. It may be shorter for an identity matrix and longer for a three-variable system that requires swaps and several eliminations.
Unique, inconsistent and dependent systems
A unique solution appears when every variable has a pivot and there is no contradiction. The large result preserves exact fractions. A separate decimal approximation uses up to twelve significant digits and is not used to verify the answer. Each original equation is checked by exact substitution.
A contradiction such as 0 = 1 means there is no solution, even if another equation looks valid on its own. A system with no contradiction but fewer pivots than variables has infinitely many solutions. The calculator gives a parameter family rather than choosing one arbitrary answer. Each parameter is an independent real number; its constant vector and all parameter directions are verified against the original equations.
For two variables, two distinct nonparallel lines meet at a unique point. Distinct parallel lines have no common point; the same line repeated has infinitely many common points. Use the Graphing Calculator to inspect nonvertical line equations visually, remembering that a graph window is not an exact algebraic proof.
Worked examples for two and three variables
For 2x + y = 15 and 3x − y = 5, adding the equations gives 5x = 20, so x = 4. Substitution into the first equation gives y = 7. Both checks succeed: 2(4) + 7 = 15 and 3(4) − 7 = 5. The calculator may perform equivalent row operations in a different order, but the verified solution is the same.
For x + y = 2 and 2x + 2y = 4, the second equation repeats the first. There is no unique pair: y can be a parameter t, and x = 2 − t. Changing the second right side to 5 creates a contradiction and no solution. In three variables, x + y + z = 6, x − y + z = 2 and 2x + y − z = 1 give x = 1, y = 2 and z = 3.
Supported equations, precision and limitations
This page accepts a square two-by-two or three-by-three linear system through its coefficient fields. Products of variables, squared variables, functions, inequalities and complex coefficients are outside its scope. Expand and rearrange an equation yourself before entering coefficients. A quadratic problem belongs in the Quadratic Formula Calculator, not in these linear fields.
Decimals may have at most six places and each coefficient value must have magnitude no greater than one million. Fraction numerators and denominators have the same magnitude limit. Exponents, commas, equations pasted into a coefficient field and empty entries are rejected. These limits keep the rational arithmetic and explanation manageable on mobile browsers.
Correct arithmetic cannot repair a wrongly transcribed equation or an unsuitable model. Check signs, units and assumptions in the original problem. A mathematically valid solution may still violate a practical constraint, such as nonnegative quantities. Calculations run locally without saving equation entries; the website's separately disclosed analytics and advertising services remain in place.
System of Equations Calculator FAQs
Sources and methodology
These educational references explain the mathematical ideas. This independently implemented calculator is not affiliated with their publisher. Its supported methods, exact checks and limits are documented above.