Three ratios, one verdict
Show on a big screenSix numbers can tell how two lines meet, before you draw anything.
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- Two equations, in the form a·x + b·y + c = 0.
- Pull out the six coefficients.
- Compare three ratios: a₁/a₂, b₁/b₂, c₁/c₂.
- Nothing is equal, so exactly one common pair.
The rule
The verdict is how the lines meet. They cross once, run parallel, or lie on one line. Write both equations as a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. Here a and b are the numbers beside x and y. The c is the number alone. The small 1 and 2 mark the first and second equation. Then compare the ratios a₁/a₂, b₁/b₂ and c₁/c₂. They need a₂, b₂ and c₂ not 0. The end of this lesson covers 0.
| The ratios say | The lines do | Common pairs | The name |
|---|---|---|---|
a₁/a₂ ≠ b₁/b₂ | cross once | exactly one | consistent |
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | run parallel | none | inconsistent |
a₁/a₂ = b₁/b₂ = c₁/c₂ | lie on one line | infinitely many | dependent, consistent |
Example
Compare 5x + 2y = 22 with 10x + 4y = 47. Write both with 0 on the right: 5x + 2y − 22 = 0 and 10x + 4y − 47 = 0. The first two ratios are 5/10 and 2/4. Both are 1/2. The third is −22/−47, which is 22/47. It is not 1/2. So the lines run parallel, and there is no solution.
Check: doubling the first gives 10x + 4y = 44. The second says 47. No x and y can make 10x + 4y equal both.
A ratio with 0 at the bottom does not exist. Compare a₁b₂ with a₂b₁ instead. Equal products mean parallel or the same line. Scale one equation to match the other's x and y numbers. Then compare the constants: same means one line, different means parallel.
Try the lab to finish this lesson.