Mathematics · Algebra
Simultaneous Equations Solver
Solve two linear equations in x and y using determinants.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- System determinant = (2 × -1) − (1 × 1) = -3.
- x determinant = -9; x = -9 ÷ -3 = 3.
- y determinant = -3; y = -3 ÷ -3 = 1.
Definition and formula
Cramer's rule replaces one coefficient column at a time and divides its determinant by the system determinant.
a₁x + b₁y = c₁; a₂x + b₂y = c₂
Worked example
For 2x + y = 7 and x − y = 2, the solution is x = 3 and y = 1.
Common mistake
A zero determinant means the equations do not have one unique intersection.
How to learn with this calculator
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Clear answers
Frequently asked questions
What does the Two-equation system do?
Solve two linear equations in x and y using determinants.
How does the Two-equation system work?
The calculator applies a₁x + b₁y = c₁; a₂x + b₂y = c₂. Cramer's rule replaces one coefficient column at a time and divides its determinant by the system determinant.
What can I learn from the Two-equation system?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed 2026-07-14. Calculations tested 2026-07-14.