Systems of Equations Solver — 2×2
Quick answer: Enter the six coefficients for two equations with two unknowns. Get x and y — or learn if the lines are parallel or identical.
Solve a system of two linear equations with two unknowns (2×2) using Cramer's Rule. Enter a₁, b₁, c₁ for equation 1 (a₁x + b₁y = c₁) and a₂, b₂, c₂ for equation 2. Instantly find x and y, or learn if the system has no solution (parallel lines) or infinite solutions (same line).
Advertisement
Last reviewed: April 2026Report an error
Solve: a₁x + b₁y = c₁ and a₂x + b₂y = c₂
Equation 1: a₁x + b₁y = c₁
Equation 2: a₂x + b₂y = c₂
Solution (x, y)
(1.75, 1.5)
x = 1.75, y = 1.5. The two lines intersect at this point.
x
1.75
y
1.5
Advertisement
How to Use This Systems of Equations Calculator
- 1Enter a₁, b₁, and c₁ for the first equation: a₁x + b₁y = c₁.
- 2Enter a₂, b₂, and c₂ for the second equation: a₂x + b₂y = c₂.
- 3Read the solution (x, y), or the result if no unique solution exists.
Frequently Asked Questions
- Cramer's Rule solves a linear system using determinants. For ax + by = e and cx + dy = f: determinant D = ad - bc. Then x = (ed - bf)/D and y = (af - ec)/D.
- The two equations represent parallel lines — they never intersect, so no (x, y) satisfies both at once. Happens when determinant = 0 and the equations are inconsistent.
- Both equations represent the same line. Every point on the line is a solution. Happens when one equation is a scalar multiple of the other.
- This calculator handles 2×2 systems. For 3×3, use Gaussian elimination or row reduction — techniques taught in linear algebra.
- Anywhere two constraints must be satisfied simultaneously: business (cost vs. revenue), physics (force equations), engineering, and economics.
Advertisement
Related Calculators
</> Embed this calculator on your website
<iframe src="https://calqpro.com/calculators/systems-of-equations-solver" width="100%" height="600" frameborder="0" title="Calqpro Calculator" loading="lazy"></iframe> <p>Powered by <a href="https://calqpro.com">Calqpro</a></p>
Advertisement