Linear Equations: Coefficient Ratios & Solution Types

📊 Table 3.1 - Interactive

Linear Equations: Coefficient Ratios & Solution Types

Table 3.1
Sl No.	Pair of lines	a₁a₂	b₁b₂	c₁c₂	Compare the ratios	Graphical representation	Algebraic interpretation
1.	x - 2y = 0 3x + 4y - 20 = 0	13	-24	0-20	a₁a₂ ≠ b₁b₂	Intersecting lines	Exactly one solution (unique)
2.	2x + 3y - 9 = 0 4x + 6y - 18 = 0	24	36	-9-18	a₁a₂ = b₁b₂ = c₁c₂	Coincident lines	Infinitely many solutions
3.	x + 2y - 4 = 0 2x + 4y - 12 = 0	12	24	-4-12	a₁a₂ = b₁b₂ ≠ c₁c₂	Parallel lines	No solution

🎮 Interactive Learning Modes

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🎯 Case 1: Intersecting Lines (Unique Solution)

Equations: x - 2y = 0 and 3x + 4y - 20 = 0

Key Insight: When the ratios of coefficients a₁a₂ ≠ b₁b₂, the lines intersect at exactly one point.

• a₁a₂ = 13
• b₁b₂ = -24 = -12
• Since 13 ≠ -12, the lines intersect
• Solution: One unique point of intersection

🔄 Case 2: Coincident Lines (Infinite Solutions)

Equations: 2x + 3y - 9 = 0 and 4x + 6y - 18 = 0

Key Insight: When all ratios are equal (a₁a₂ = b₁b₂ = c₁c₂), the equations represent the same line.

• a₁a₂ = 24 = 12
• b₁b₂ = 36 = 12
• c₁c₂ = -9-18 = 12
• Since all ratios are equal, lines are identical
• Solution: Infinitely many points satisfy both equations

↔️ Case 3: Parallel Lines (No Solution)

Equations: x + 2y - 4 = 0 and 2x + 4y - 12 = 0

Key Insight: When a₁a₂ = b₁b₂ but c₁c₂ is different, the lines are parallel but distinct.

• a₁a₂ = 12
• b₁b₂ = 24 = 12
• c₁c₂ = -4-12 = 13
• Since 12 = 12 but 12 ≠ 13, lines are parallel
• Solution: No point of intersection exists

🔑 Key Summary:

• a₁a₂ ≠ b₁b₂ → Intersecting lines → Unique solution
• a₁a₂ = b₁b₂ = c₁c₂ → Coincident lines → Infinite solutions
• a₁a₂ = b₁b₂ ≠ c₁c₂ → Parallel lines → No solution