What Is a 2×2 Linear System?
A 2×2 linear system is a set of two linear equations with two unknowns (x and y). The solution is the ordered pair (x, y) that satisfies both equations simultaneously.
{ a₁x + b₁y = c₁
{ a₂x + b₂y = c₂
Determinant: D = a₁b₂ − a₂b₁
x = (c₁b₂ − c₂b₁) ÷ D
y = (a₁c₂ − a₂c₁) ÷ DClassification of the system
- D ≠ 0: unique solution — the lines cross at one point
- D = 0 and Dx = Dy = 0: infinitely many solutions — coincident lines
- D = 0 and Dx ≠ 0 or Dy ≠ 0: no solution — parallel lines
Elimination (addition) method
Example: x + y = 5 and x − y = 1 Add the equations: 2x = 6 → x = 3 Substitute: 3 + y = 5 → y = 2 Solution: (3, 2)
Frequently asked questions
Can I use the substitution method?
Yes! Isolate one unknown in one equation and substitute it into the other. The calculator uses determinants (Cramer's rule), a general method for 2×2 systems.
What does the solution mean geometrically?
Each linear equation is a line in the plane. The solution of the system is the intersection point of those lines. Unique = lines cross; dependent = same line; inconsistent = parallel lines.
What if the solution is a fraction?
That is perfectly normal! The solution can be any real number, including fractions and decimals. The calculator shows the result with up to 4 decimal places.
How do I check the solution?
Substitute (x, y) into both original equations. Both must be true. If one does not hold, there was an error in the coefficients entered.