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System of equations calculator (2x2) step by step

Solve a system of two equations with two unknowns (x and y) instantly and step by step. Enter the coefficients and the calculator does the rest.

a\u2081·x + b\u2081·y = c\u2081

a\u2082·x + b\u2082·y = c\u2082

Practice exercises on sistemas de ecuaciones

How how to solve a 2x2 system

A 2x2 system consists of two equations a·x + b·y = c that must be true at the same time. This calculator uses the determinant method (Cramer's rule): the main determinant D = a₁·b₂ − a₂·b₁ is calculated and, if it is not zero, there is a unique solution.

If D = 0, the system does not have a unique solution (it is inconsistent or has infinitely many solutions).

ExampleSolve x + y = 10 ; 2x − y = 2.

Step-by-step solution

  1. Adding the two equations eliminates y: 3x = 12 → x = 4.
  2. Substituting into the first one: 4 + y = 10 → y = 6.
Solution: x = 4, y = 6.

Don't just focus on the result: learn how to do it

With Refuerzo.ai you practice systems of equations with hundreds of exercises and instant feedback that explains why. Try for free, no card required.

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Frequently asked questions

What methods are there to solve a 2x2 system?

Substitution, comparison, and elimination give the same result. This calculator uses determinants (Cramer's rule), which is fast and systematic.

When does a system have no solution?

When the main determinant is zero: the system is inconsistent (parallel lines, no solution) or dependent (the same line, infinitely many solutions).

Is the calculator free?

Yes, the systems of equations calculator is free. And if you want to practice with hundreds of instantly corrected exercises, you have a 5-day trial with no card required on Refuerzo.ai.