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.
LaTeX article Updated July 13, 2026
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.
Worked example
Common mistakes
- Calling 0 = 0 one ordered-pair solution.
- Calling 0 = 6 an arithmetic mistake automatically.
- Comparing only one pair of coefficients.
Keep these ideas
- Different slopes meet once.
- Parallel distinct lines never meet.
- Identical equations share a whole line of solutions.