How to Solve Systems of Equations
A system of equations is a set of equations that share the same variables — the solution is the point (x, y, …) satisfying all of them at once. Two straight lines usually cross at exactly one point; that point is the solution.
The Main Methods
- Elimination: add or subtract multiples of equations to cancel one variable at a time — the most mechanical method, used for the 3×3 solver above.
- Substitution: solve one equation for a variable and plug it into the others — great for 2×2 by hand.
- Cramer’s rule: express each variable as a ratio of determinants — elegant for 2×2, shown step by step above.
Special Cases
If elimination produces 0 = 5, the system is inconsistent (no solution — parallel lines). If it produces 0 = 0, the equations are dependent (infinitely many solutions — the same line twice). The calculator detects and reports both.
Working with single equations or matrices? Try our linear equation solver or the matrix calculator.