N Numeratibase

Free math tool

Systems Of Equations Solver

Solve a 2x2 linear system in x and y, show the determinant-based reasoning, and identify parallel or identical lines.

Reset
Unique solution No solution Same line

Direct answer

System solution

The solution is (3, 2).

x 3
y 2

Steps

  1. Write the system in standard form: x + y = 5 and x - y = 1.
  2. Compute the determinant: (1)(-1) - (1)(1) = -2.
  3. Solve for x and y to get x = 3 and y = 2.

Why it works

A system solution is the point where both equations are true at once. For two linear equations, that means finding where the two lines intersect or recognizing when they are parallel or identical.

Common mistakes

  • Comparing only coefficients and forgetting to compare constants too.
  • Solving each equation separately instead of solving them together.
  • Missing that proportional equations can mean no solution or infinitely many solutions depending on the constants.

Worked examples

Solve x + y = 5 and x - y = 1

The intersection point is (3, 2).

Solve x + y = 2 and 2x + 2y = 8

The lines are parallel, so there is no solution.

Practice

  • x + 2y = 8 and x - y = 2
  • 2x + y = 7 and 4x + 2y = 14
  • x + y = 3 and 2x + 2y = 10