Interactive Linear Equations Explorer
Understand the nature of solutions for a pair of linear equations.
Enter Your Equations
Enter the coefficients for two linear equations in the form ax + by + c = 0.
Equation 1: 1x + 3y + -6 = 0
x +
y +
= 0
Equation 2: 2x + -3y + -12 = 0
x +
y +
= 0
Analysis
Ratios
a₁a₂=
b₁b₂=
c₁c₂=
Comparison
Graphical Representation
Intersecting Lines
Algebraic Interpretation
Exactly one solution (unique)
Graphical Representation
Understanding the Conditions
Intersecting Lines
a₁a₂
≠
b₁b₂
The lines cross at exactly one point. This means there is a single, unique pair of (x, y) values that satisfies both equations.
Exactly One Solution
Coincident Lines
a₁a₂
=
b₁b₂
=
c₁c₂
The lines are identical and overlap completely. Every point on the line is a solution to both equations.
Infinitely Many Solutions
Parallel Lines
a₁a₂
=
b₁b₂
≠
c₁c₂
The lines never cross. They have the same slope but different y-intercepts. There is no (x, y) pair that satisfies both equations.
No Solution