Overview
A 2×2 linear system describes two straight lines in the plane. The solution is the point where the lines intersect. Parallel lines have no solution, while identical lines share every point as a solution.
Formula
a₁x + b₁y = c₁ and a₂x + b₂y = c₂. The determinant is D = a₁b₂ − a₂b₁, then x = Dₓ/D and y = D_y/D.
How it works
Arrange both equations
Write each equation in the form ax + by = c.
Compute the determinant
D = a₁b₂ − a₂b₁. If it is zero, the system has no unique solution.
Compute Dₓ and D_y
Replace the coefficient column with the constants column.
Divide to get the solution
x = Dₓ/D and y = D_y/D, then verify by substitution.
Worked examples
Example 1: 2x + y = 7 and x − y = 2
D = −3, Dₓ = −9, D_y = −3, so x = 3 and y = 1.
Example 2: x + y = 2 and 2x + 2y = 4
The determinant is zero and the equations are identical, so there are infinitely many solutions.
Frequently asked questions
What does a zero determinant mean?
The coefficients are proportional, so there is no unique solution; there is either no solution or infinitely many.
Can I use fractions?
Yes. Enter them as decimals such as 0.5, or convert them with the fraction calculator first.
Why verify by substitution?
To confirm the solution satisfies both equations, not just one.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.