The pair, the picture, the verdict
A linear equation in two letters draws one straight line.
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x + 2y = 6, drawn as a line.- One equation,
x + 2y = 6, drawn as a line. - A parallel line,
x + 2y = 8. They never meet. - The same claim again,
2x + 4y = 12. Every point is shared. - A different lean,
x − y = 0. They cross once, at (2, 2).
The rule
In a linear equation, each letter has power 1. That means x, not x². x + 2y = 6 is linear. A solution is a pair of numbers, one for each letter. The pair makes the equation true. It is a point on the line.
Two lines can cross once, run parallel, or lie on one line. So a pair of equations has one, none, or infinitely many solutions. It cannot have exactly two. Two different lines meet at most once.
| Second equation | The lines do | Solutions of the pair | The name |
|---|---|---|---|
x + 2y = 8 | run parallel | none | inconsistent |
2x + 4y = 12 | lie on one line | every point of it | dependent |
x − y = 0 | cross once | exactly one, (2, 2) | consistent |
A pair with at least one solution is consistent. A pair with none is inconsistent. A pair whose two equations draw one line is dependent. It has solutions, so it is consistent too.
Watch out!
A crossing off your sheet is still a crossing. Lines that are not parallel must meet. The lines x + y = 12 and x − y = 2 meet at (7, 5).