One solution, no solution, or infinitely many?
For two linear equations, the coefficient pattern determines whether the graphs intersect once, never meet, or are actually the same line.
What the idea means, why its conditions matter, and where it connects.
Updated July 13, 2026A consistent independent system has one solution, an inconsistent system has none, and a dependent system has infinitely many.
One solution means different slopes
Two nonvertical lines with different slopes cross exactly once. Algebraically, elimination or substitution produces a definite x and y.
A vertical line and a nonvertical line also meet once unless a domain restriction says otherwise.
No solution means parallel and distinct
If the variable coefficients are proportional but the constants are not in the same ratio, the lines have the same direction and different positions.
Elimination reduces the system to a false statement such as 0 = 6.
Infinitely many means one equation is a rewrite of the other
When all coefficients and constants are proportional, both equations name the same line. Every point on that line satisfies both.
Elimination produces a true identity such as 0 = 0. The solution should be described as the full line, not as every point in the plane.
Classify 4x − 6y = 8 and 2x − 3y = 5.
Scale the second equation to compare coefficients.
The left sides match but the constants differ.
Subtracting produces a contradiction.
Common mistakes
- Calling 0 = 0 one ordered-pair solution.
- Calling 0 = 6 an arithmetic mistake automatically.
- Comparing only one pair of coefficients.
Three takeaways
- Different slopes meet once.
- Parallel distinct lines never meet.
- Identical equations share a whole line of solutions.