Simultaneous Equations Solver – 2×2 Systems

A 2×2 linear system intersects two lines at a single point (usually). The solver uses elimination and prints every step for the system 2x + y = 5, x − y = 1: adding the equations removes y (3x = 6), giving x = 2, and back-substitution into either equation returns y = 1. Verification runs both ways: 2(2)+1 = 5 and 2−1 = 1. The determinant D = a₁b₂ − a₂b₁ = 2(−1) − 1(1) = −3 ≠ 0 certifies a unique solution; when D = 0 the lines are parallel (no solution) or identical (infinitely many), and the tool reports which. Substitution and Cramer’s rule solutions appear alongside, so the answer can be cross-checked three ways. Applications include break-even analysis, mixture problems, and Kirchhoff circuit pairs.

1. 1

   Enter both equations

   Coefficients a, b, c for each of the two lines

2. 2

   Check the determinant

   D = a₁b₂ − a₂b₁ — nonzero guarantees one solution

3. 3

   Eliminate and back-substitute

   Each arithmetic step is displayed

4. 4

   Verify in both equations

   The solution is substituted into each original line

• 1

  D ≠ 0 ⇒ unique intersection; D = 0 ⇒ parallel or identical lines

• 2

  Elimination targets the variable with equal-opposite coefficients

• 3

  Always substitute the answer back into BOTH equations

• Sign slips when adding equations to eliminate a variable

• Solving for x and forgetting to back-substitute for y

• Missing the D = 0 case and reporting a phantom solution

• Mixing coefficient order between the two equations