The three things two lines can do
Show on a big screenKeep line 1. Change line 2. Watch what the two lines do.
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2x + 3y = 12.- 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.
Try the lab to finish this lesson.