System of Equations Calculator
Solve a pair of linear equations in x and y using Cramer's rule, including the no-solution and infinite-solution cases.
Equation 1: a₁x + b₁y = c₁
Equation 2: a₂x + b₂y = c₂
Cramer's rule
- D = a₁b₂ − a₂b₁ = 6
- Dₓ = c₁b₂ − c₂b₁ = 9
- D_y = a₁c₂ − a₂c₁ = 6
- x = Dₓ / D = 1.5
- y = D_y / D = 1
How it works
A 2×2 system is two straight-line equations, a₁x + b₁y = c₁ and a₂x + b₂y = c₂, and solving it means finding the point where the lines cross. This calculator uses Cramer's rule, which handles it with three determinants.
First it computes the main determinant D = a₁b₂ − a₂b₁. If D isn't zero, there's exactly one crossing point, found by x = Dₓ / D and y = D_y / D, where Dₓ and D_y swap the c-column into the x- and y-columns.
If D is zero, the lines are parallel. Either they're the very same line — infinitely many solutions — or they never meet at all — no solution. The tool checks the other determinants to tell you which case you've got.
Frequently asked questions
What does a determinant of zero tell me?
It means the two lines have the same slope, so they're parallel. If they're also the same line, every point solves the system (infinitely many solutions). If they're distinct parallel lines, they never cross, so there's no solution.
What is Cramer's rule?
It's a formula that solves a linear system using determinants instead of substitution or elimination. For a 2×2 system you compute three 2×2 determinants and divide, which is fast and easy to check by hand.
Can it handle decimals and negative coefficients?
Yes. Any real numbers work for the coefficients and constants. The arithmetic is the same, so decimals like 1.5 or negatives like −3 are fine.