Concept 3.2
Reading the Ratios
Source: NCERT Mathematics, Class 1010 minute lesson
Formal statement
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: if a₁/a₂ ≠ b₁/b₂ the lines intersect (unique solution); if a₁/a₂ = b₁/b₂ ≠ c₁/c₂ they are parallel (no solution); if a₁/a₂ = b₁/b₂ = c₁/c₂ they coincide (infinitely many solutions).
The intuition
The ratios a₁/a₂ and b₁/b₂ compare the directions of the two lines. If they differ, the lines point different ways and must cross.
Only when the directions match does c₁/c₂ matter: it tells you whether you have the same line twice or two parallel copies.
The attention
Look at a₁/a₂ and b₁/b₂ first. c₁/c₂ only matters if they match.
a₁/a₂ vs b₁/b₂ first, then c₁/c₂
Look hereNot here
- Look here
- a₁/a₂ and b₁/b₂, the directions. These two ratios decide whether the lines point the same way. Different directions mean exactly one crossing, whatever the constants are.
- Not here
- c₁/c₂, the position. c₁/c₂ is about where a line sits, not which way it points. Checked first, it can't tell you anything on its own.
- What it teaches
- Ask the deciding question first. Some details only matter after it is answered.