System of Equations Solver
Solve a 2×2 linear system a₁x + b₁y = c₁, a₂x + b₂y = c₂ using Cramer’s rule.
Equation 1: a₁x + b₁y = c₁
Equation 2: a₂x + b₂y = c₂
How it works
- 1Enter the six coefficients a₁, b₁, c₁ and a₂, b₂, c₂ for the two linear equations.
- 2The solver computes the determinant D = a₁b₂ − a₂b₁. When D ≠ 0, Cramer’s rule gives x = (c₁b₂ − c₂b₁) ÷ D and y = (a₁c₂ − a₂c₁) ÷ D.
- 3When D = 0 the system is either inconsistent (parallel lines, no solution) or dependent (same line, infinitely many solutions).
Use cases
- Checking homework answers for systems of linear equations in algebra class.
- Engineering and physics problems where two constraints give two simultaneous equations.
- Quickly finding the intersection point of two lines given in standard form.
Frequently asked questions
What does it mean when the determinant is zero?
A zero determinant means the two lines are either parallel (no intersection, no solution) or identical (infinitely many solutions). The solver distinguishes the two by checking whether the equations are proportional.
Can I enter decimal or negative coefficients?
Yes — all six fields accept any real number, including decimals and negatives. Results are shown to four decimal places.
What is Cramer’s rule?
Cramer’s rule expresses each unknown as a ratio of determinants. For a 2×2 system, x and y are found by replacing the relevant coefficient column with the constants and dividing by the coefficient determinant.
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