Simultaneous Equations Calculator
Solve two linear equations in x and y from full equation input. Supports parentheses, fractions, Cramer's rule steps, graph, and substitution check.
Two equations with two unknowns: enter full equations, let the calculator collect coefficients, solve by Cramer's rule, and verify the result by substitution.
Enter two equations directly
English search intent for a simultaneous equations calculator usually expects full equation input, not only coefficient boxes. This page accepts two linear equations in x and y with variables on either side of the equals sign.
x and y are the unknowns; a1, b1, c1, a2, b2, and c2 are collected after each equation is simplified.
The parser expands parentheses, reads numeric fractions, normalizes decimals, moves all terms to one side, and reduces each equation to a two-variable linear row.
If you already have a coefficient matrix, use the 2x2 system solver. This page is for pasting the original equations and seeing how the calculator classifies the result.
Cramer's rule steps
For a two-variable linear system, the calculator computes the main determinant and two replacement determinants. Substitution and elimination lead to the same solution, but the displayed algebra path uses Cramer's rule.
Delta is the main determinant; it shows whether the two lines have a unique intersection point.
Delta x and Delta y are the replacement determinants for x and y.
When Delta is not zero, divide the replacement determinants by the main determinant.
A nonzero main determinant means the system has exactly one ordered pair (x, y).
Possible outcomes
| Delta | Delta x, Delta y | Result |
|---|---|---|
| not 0 | - | One solution (x, y) |
| 0 | both 0 | Infinitely many solutions |
| 0 | at least one nonzero | No solution |
When the system has infinitely many solutions, the calculator identifies the case but does not write a full parametric solution. For that, choose a free parameter and express one variable through the other.
Supported input
| Supported | Not supported |
|---|---|
| 2x + 3y = 8 | x² + |
| - 1 | |
| + | 1/x + |
| 3(x - 1) + 2y = 5 | sin(x) + |
Only linear equations in x and y are supported. Squares, products of variables, division by a variable, or functions of variables make the system nonlinear and outside this calculator's model.
Example results
| System | Outcome |
|---|---|
| x +; x - | one solution |
| 3x - 2y = 4; 6x - 4y = 8 | infinitely many solutions |
| x +; x + | no solution |
Frequently Asked Questions
Sources and References
- System of EquationsWolfram MathWorld
- Cramer's RuleWolfram MathWorld
- 2x2 System of Equations SolverMath Portal
Calculations are based on the listed reference sources. Links open in a new tab.
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