System of Equations Calculator
Enter two linear equations below to instantly solve for x and y.
a₁x + b₁y = c₁ and a₂x + b₂y = c₂
Solving with Cramer's Rule
For a₁x + b₁y = c₁ and a₂x + b₂y = c₂: x = (c₁b₂ − c₂b₁) / (a₁b₂ − a₂b₁), and y = (a₁c₂ − a₂c₁) / (a₁b₂ − a₂b₁).
Worked example
2x + y = 10 and x − y = 2.
Determinant = (2)(−1) − (1)(1) = −3
x = (10(−1) − 2(1)) / −3 = 4, y = (2(2) − 1(10)) / −3 = 2
Solution: x = 4, y = 2
Keep going
- Solving a system means finding where two lines intersect; the linear equation calculator solves just one of those lines for x on its own. Linear Equation Calculator
- A 2x2 system of equations can also be solved with matrix methods; the matrix calculator handles the determinant and inverse math behind that approach. Matrix Calculator
Frequently Asked Questions
What is a system of equations?
A system of equations is a set of two or more equations sharing the same variables. Solving it means finding values for those variables that satisfy every equation at once.
What does it mean geometrically?
Each linear equation represents a line. The solution to the system is the point where the lines intersect. If the lines are parallel, there's no solution; if they're the same line, there are infinitely many.
What method does this calculator use?
Cramer's rule, which uses determinants to solve directly for x and y without needing to substitute one equation into the other by hand.
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