Math Solver Free

System of Equations Solver

Solve 2ร—2 and 3ร—3 linear systems โ€” every elimination step shown.

x+y=
x+y=
x =
2.2
y =
1.2

Step 1: D = aโ‚bโ‚‚ โˆ’ aโ‚‚bโ‚ = (2)(-1) โˆ’ (1)(3) = -5

Step 2: Dx = cโ‚bโ‚‚ โˆ’ cโ‚‚bโ‚ = -11, Dy = aโ‚cโ‚‚ โˆ’ aโ‚‚cโ‚ = -6

Step 3: x = Dx/D = 2.2, y = Dy/D = 1.2

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How to Solve Systems of Equations

A system of equations is a set of equations that share the same variables โ€” the solution is the point (x, y, โ€ฆ) 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 โ€” the most mechanical method, used for the 3ร—3 solver above.
  • Substitution: solve one equation for a variable and plug it into the others โ€” great for 2ร—2 by hand.
  • Cramerโ€™s rule: express each variable as a ratio of determinants โ€” elegant for 2ร—2, shown step by step above.

Special Cases

If elimination produces 0 = 5, the system is inconsistent (no solution โ€” parallel lines). If it produces 0 = 0, the equations are dependent (infinitely many solutions โ€” 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.

Frequently Asked Questions

What methods are used to solve the system?

For 2x2 systems the calculator uses Cramer's rule (determinants); for 3x3 systems it uses Gaussian elimination with back-substitution. Every step is shown so you can follow the working.

What does it mean when there is no solution?

No solution means the equations are inconsistent โ€” geometrically, the lines (or planes) are parallel and never meet. If the equations describe the same line, there are infinitely many solutions instead.

Can I solve systems with more than 3 variables?

This tool handles 2x2 and 3x3 systems, which cover nearly all school and college coursework. For larger systems, the matrix calculator on this site can help with the underlying matrix operations.

Is this system of equations solver free?

Yes โ€” every tool on Math Solver Free is free with no sign-up, and all calculations run in your browser.