What the solutions mean
What this lesson covers
- Parallel lines give no solution.
- Identical lines give infinitely many solutions.
- A negative discriminant means a line misses a curve.
Three cases for two lines
- Compare the equations.
| System | What happens | Solutions |
|---|---|---|
| 2x + y = 5 and x − y = 1 | Cross once | One |
| 2x + y = 5 and 4x + 2y = 3 | Parallel | None |
| 2x + y = 5 and 4x + 2y = 10 | Same line | Infinitely many |
A line that misses
- y = x − 3 and y = x².
- x² − x + 3 = 0
- b² − 4ac = 1 − 12 = −11
- No real solutions, so they do not meet
Common mistakes
- Saying 0 = 7 means you made a mistake.It can be correct: it shows there is no solution.
- Saying 0 = 0 means x = 0.It means the equations are the same line.
Your turn
- 1. How many solutions: x + 2y = 4 and 2x + 4y = 8?
- 2. How many solutions: 3x − y = 2 and 6x − 2y = 5?
Show answer
1. Infinitely many, as they are the same line 2. None, as they are parallel
What to take away
- One, none or infinitely many: always say which.
- Link the answer to the graph.
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