Use the standard form: a·x + b·y = c. You can enter negative values and fractions.
Example data table
| Equation 1 | Equation 2 | Solution (x, y) |
|---|---|---|
| 2x + 3y = 13 | x − y = 1 | (3.2, 2.2) |
| x + y = 10 | 2x − y = 5 | (5, 5) |
| 3x − 6y = 9 | x − 2y = 3 | Infinite solutions |
Examples are provided for learning and quick testing.
Formula used
This calculator solves a 2×2 linear system using determinants (Cramer’s rule):
- D = a1·b2 − a2·b1
- Dx = c1·b2 − c2·b1
- Dy = a1·c2 − a2·c1
- If D ≠ 0: x = Dx/D and y = Dy/D
If D = 0, the system has no unique solution. When Dx = 0 and Dy = 0, solutions are infinite. Otherwise, no solution exists.
How to use this calculator
- Write both equations in the form a·x + b·y = c.
- Enter a1, b1, c1 for the first equation.
- Enter a2, b2, c2 for the second equation.
- Select output format and rounding preference.
- Press Solve now to view results above the form.
- Use export buttons to save CSV or PDF reports.
For best accuracy, use fraction output when possible.
FAQs
1) What inputs does the solver accept?
You can enter integers, decimals, or fractions like 3/4. Negative values are allowed. Keep numbers reasonably sized for reliable exact fraction results.
2) What does “D” mean?
D is the determinant of the coefficient matrix. If D is not zero, a single unique intersection point exists and the solver can compute x and y directly.
3) Why do I see “no solution”?
No solution appears when the lines are parallel and never intersect. Mathematically, this happens when D equals zero but Dx or Dy is not zero.
4) What does “infinite solutions” mean?
It means both equations describe the same line. Every point on that line satisfies both equations, so there is no single unique pair (x, y).
5) Should I use decimal or fraction output?
Use fractions for exact results when inputs are fractions or integers. Use decimals when you want rounded values, especially for reporting or quick estimation.
6) How does verification help?
Verification substitutes the computed x and y back into both equations. If the left sides match the targets, your solution is consistent. Small differences can happen after rounding.
7) Can this solve non-linear equations?
This tool is designed for linear equations in two variables. For non-linear systems, you typically need different methods such as substitution with factoring or numerical solvers.
8) What if one equation has missing x or y?
Enter 0 for the missing variable’s coefficient. For example, 0x + 5y = 10 is valid and will be handled correctly by the determinant checks.