The three things two lines can do
Keep line 1. Change line 2. Watch what the two lines do.
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2x + 3y = 12. Step 1 of 4
- Line 1:
2x + 3y = 12. - Line 2 as
2x + 3y = 18. Same lean, never meets line 1. - Line 2 as
4x + 6y = 24. The very same line as line 1. - Line 2 as
3x − 4y = 5. A different lean, crosses once.
The rule
Rule: Two lines cross once, never, or lie on one line. A pair of equations has exactly one solution, none, or infinitely many. It never has exactly two.
A verdict names which of these three cases applies. A line's lean is its steepness. Parallel lines have the same lean and never meet.
| Second line | The two lines | Solutions of the pair |
|---|---|---|
2x + 3y = 18 | run parallel | none |
4x + 6y = 24 | are one line | every point of it |
3x − 4y = 5 | cross once | exactly one |
A pair of equations with a solution is consistent. A pair with none is inconsistent. A pair whose two equations draw one line is dependent. A dependent pair is also consistent.
One trap looks like a doubled pair. Take 4x + 6y = 25 as line 2. Every coefficient is double line 1's. The lean matches. Because 25 is not double 12, the lines never meet. This pair looks dependent but is inconsistent.