Enter equations and read the result
Put each simultaneous linear equation on its own line in the Linear equations (one per line, 2 or more) field. For 2x + y = 5 and x - y = 1, Systems of Equations Calculator uses Gaussian elimination, keeps exact values, and adds a decimal approximation when a unique value is not an integer.
Use Systems of Equations Calculator
- Type each equation in the Linear equations (one per line, 2 or more) field.
- Click Solve to parse the lines and calculate the system.
- Click Load example to fill in
2x + y = 5andx - y = 1. Click Reset to clear the input and result. - After a result appears, click Copy result to copy the equations, solution, and row-operation steps, or click Download result (.txt) to save that output as a text file.
Which equations can Systems of Equations Calculator read?
Enter one equation per line, with exactly one = sign on each line. The parser accepts numbers, spaces, decimal points, parentheses, lowercase single-letter variables, and + - * / ^. It also understands implicit multiplication, so 2x and 3(x + 1) work.
- Use 2 to 8 nonblank equations, with no more than 8 distinct variables.
- Keep each line at 200 characters or fewer.
- Use a numeric denominator, such as
x/2or1/2x; a variable in a denominator is rejected. - Keep terms linear. Products such as
xy, variable powers such asx^2, negative or fractional exponents, and division by zero produce errors.
Read the Gaussian elimination result
The calculator forms Ax = b and reduces the augmented matrix [A | b] with Gaussian elimination and exact fraction arithmetic. The result panel lists the row operations and reports one solution, infinitely many solutions with free variables, or no solution for an inconsistent system.
2x + y = 5
x - y = 1
x = 2
y = 1For a unique non-integer solution, the displayed variable value includes the exact fraction and a decimal approximation. Copy result and Download result (.txt) include the entered equations, solution, and calculation steps.
Check free variables and inconsistent systems
A system with a free variable produces expressions for the pivot variables and identifies the free variable. When elimination produces a contradiction, the result labels the system as inconsistent and reports no solution.
Choose a calculator for the equation you have
Use Math Calculators to browse the category. Choose Linear Equation Calculator for a one-variable linear equation. Choose Matrix Calculator for matrix addition, multiplication, transpose, determinant, inverse, rank, or solving Ax = b. Choose Algebra Calculator for a single-variable quadratic, a small linear system, or another supported algebra task.
Frequently asked questions
How do I solve simultaneous linear equations here?
Enter two or more equations in the Linear equations (one per line, 2 or more) field and click Solve. The result shows the solution status and Gaussian elimination steps.
Can I enter fractions and decimals in a system of equations?
Yes. Numeric fractions and decimals are accepted when the denominator is numeric. Enter forms such as 1/2x or 0.5x; a variable denominator is rejected.
What does no solution mean in the calculator result?
No solution means the equations are inconsistent. Gaussian elimination has produced a contradiction, so no set of variable values satisfies every entered equation.
Why does my equation produce an input error?
Errors come from malformed lines, unsupported characters, missing sides of an equals sign, unbalanced parentheses, nonlinear terms, variable denominators, division by zero, or more than 8 equations or variables. Each nonblank line must describe a supported linear equation.
Are my equations processed in the browser?
Yes. The calculator parses and calculates the equations in your browser. Its calculation script does not upload the equations or execute the input as code.