The ratios tell you first
Compare the ratios of the coefficients to know how the lines sit.
Look before you draw
Here are two equations: x + 2y − 4 = 0 and 2x + 4y − 12 = 0. Without drawing anything…
How do their lines sit?
What this lesson covers
The idea
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂ gives intersecting lines, a₁/a₂ = b₁/b₂ = c₁/c₂ coincident lines, a₁/a₂ = b₁/b₂ ≠ c₁/c₂ parallel lines, and conversely.
Look before you draw
Here are two equations: x + 2y − 4 = 0 and 2x + 4y − 12 = 0. Without drawing anything…
How do their lines sit?
- They cross
- They are parallel
- They are the same line
Read the ratios
Each card shows the ratios of matching coefficients: a₁/a₂, b₁/b₂ and c₁/c₂. Use them to say how the two lines sit. The picture checks you.
The rule
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: a₁/a₂ ≠ b₁/b₂: intersecting lines a₁/a₂ = b₁/b₂ = c₁/c₂: coincident lines a₁/a₂ = b₁/b₂ ≠ c₁/c₂: parallel lines
The converse is also true: if the lines intersect, then a₁/a₂ ≠ b₁/b₂, and so on. Each picture has exactly one set of ratios.
Why? When a₁/a₂ = b₁/b₂, the x and y parts of the first equation are the same multiple of those of the second, so both lines run the same way. Then c₁/c₂ decides: equal means the same line, different means a shifted copy.
Notes
a₁/a₂ ≠ b₁/b₂: intersecting a₁/a₂ = b₁/b₂ = c₁/c₂: coincident a₁/a₂ = b₁/b₂ ≠ c₁/c₂: parallel
Check yourself
For 3x + 2y − 6 = 0 and 6x + 4y − k = 0, the lines coincide when k = ?
Answer: 12
We need −6/(−k) = 6/k = 1/2, so k = 12. Then the second equation is 2 times the first.
2x + 5y − 3 = 0 and 4x + 10y − 5 = 0: the ratios are 1/2, 1/2 and 3/5. The lines are…
x + y − 5 = 0 and 2x − y − 4 = 0: a₁/a₂ = 1/2 and b₁/b₂ = −1. How do the lines sit?
If a₁/a₂ = b₁/b₂ = c₁/c₂, the pair has…
- parallel, with no solution — correct. Yes! a₁/a₂ = b₁/b₂ but c₁/c₂ is different.
- coincident. Coincident lines need c₁/c₂ = 1/2 as well, and here it is 3/5.
- intersecting. Intersecting lines need a₁/a₂ ≠ b₁/b₂. Here they are equal.
- They intersect at one point — correct. Yes! a₁/a₂ ≠ b₁/b₂. (They meet at (3, 2).)
- They are parallel. Parallel lines need a₁/a₂ = b₁/b₂.
- They are the same line. The same line needs all three ratios equal.
- infinitely many solutions: the lines coincide — correct. Yes! The pair is dependent and consistent.
- no solution. That needs c₁/c₂ to be different from the others.
- exactly one solution. That needs a₁/a₂ ≠ b₁/b₂.