What is an inconsistent system?

A system with no solution, which occurs when the two lines are parallel (same slope, different intercepts). The determinant is zero and the equations contradict each other — no (x, y) pair satisfies both simultaneously.

What is a dependent system?

A system with infinitely many solutions, where both equations represent the same line. The determinant is zero but the equations are multiples of each other — every point on the line is a solution.

What is Cramer's Rule?

A method using 2×2 determinants to solve linear systems. For the system a₁x + b₁y = c₁, a₂x + b₂y = c₂: x = det([c₁,b₁; c₂,b₂]) / det([a₁,b₁; a₂,b₂]). It only works when the coefficient matrix has a non-zero determinant.

How do I check if my solution is correct?

Substitute the values of x and y back into both original equations. Both sides of each equation should be equal. The calculator shows this verification automatically for Equation 1.

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IPL 2026

System of Equations Calculator

Solve a system of two linear equations with two unknowns (2×2). Finds x and y using Cramer's Rule and shows the verification. Useful for algebra homework, economics models, mixture problems, and any scenario with two simultaneous constraints.

Complete User Guide

Enter the coefficients a₁, b₁, and constant c₁ for Equation 1 (a₁x + b₁y = c₁), and a₂, b₂, c₂ for Equation 2. The calculator computes the determinant and applies Cramer's Rule to find the unique intersection point (x, y).

The Mathematical Formula

a₁x + b₁y = c₁, a₂x + b₂y = c₂

Using Cramer's Rule: det = a₁·b₂ − a₂·b₁. Then x = (c₁·b₂ − c₂·b₁) / det and y = (a₁·c₂ − a₂·c₁) / det. If det = 0, the system is either inconsistent (parallel lines, no solution) or dependent (same line, infinite solutions).

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