Enter any two linear equations in Ax + By = C form to instantly determine whether the system has one solution (lines intersect), no solution (parallel lines), or infinite solutions (the same line) — with a visual diagram and step-by-step slope analysis.
a₁x + b₁y = c₁
a₂x + b₂y = c₂
Every system of two linear equations falls into exactly one of three categories based on how many times the lines cross:
Consistent (One Solution) — The two lines have different slopes, so they intersect at exactly one point. That point is the solution.
Inconsistent (No Solution) — The two lines have the same slope but different y-intercepts. They are parallel and never meet. The system is called inconsistent. Solution set: ∅
Dependent (Infinite Solutions) — Both equations describe the exact same line. Every point on the line satisfies both equations. The system is called dependent.
Convert to y = mx + b, then compare m and b.
The fastest way to classify any system is to convert both equations to slope-intercept form (y = mx + b) and compare:
Determinant shortcut: Compute det = a₁b₂ − a₂b₁. If det ≠ 0, there is exactly one solution. If det = 0, check whether the right-hand sides are proportional to decide between no solution and infinite solutions.
A focused tutoring session makes these three cases crystal-clear. Work through real exam problems with personalized, step-by-step guidance — no confusion left behind.