Chapter 3: Pair of Linear Equations
Decoding the graphical and algebraic behavior of intersecting, parallel, and coincident lines.
📊 The Ultimate Condition Matrix
Comparing ratios for lines: a₁x + b₁y + c₁ = 0 & a₂x + b₂y + c₂ = 0
| Compare Ratios | Graphical View | Algebraic Solution | Consistency |
|---|---|---|---|
|
a₁a₂ ≠
b₁b₂
|
Intersecting lines |
Exactly one solution (Unique) |
Consistent |
|
a₁a₂ =
b₁b₂ =
c₁c₂
|
Coincident lines |
Infinitely many solutions |
Dependent (Consistent) |
|
a₁a₂ =
b₁b₂ ≠
c₁c₂
|
Parallel lines |
No solution | Inconsistent |
⚙️ Algebraic Methods
Step 1: Find the value of one variable (e.g., y) in terms of the other (x) from one equation.
Step 2: Substitute this into the second equation to solve for x.
Step 1: Multiply equations by constants to make coefficients of one variable numerically equal.
Step 2: Add or subtract the equations to eliminate that variable and solve the resulting single-variable equation.
⚠️ Common Exam Traps
Trap 1: The "True" Anomaly
While substituting or eliminating, if you get a true statement with NO variables (like 18 = 18), don't panic! It means the original pair has infinitely many solutions.
Trap 2: The "False" Anomaly
If you obtain a mathematically false statement involving no variables (like -4 = 0), the equations are inconsistent and have no solution (the lines are parallel).
⭐ Board Exam Super-Hacks
Standard Form is King
Always bring equations to the standard form ax + by + c = 0 before extracting a₁, b₁, c₁ to compare ratios. A common mistake is leaving constants on opposite sides of the equals sign.
The "Dependent" Trick
If an MCQ asks for a "dependent" pair, it simply means "infinitely many solutions" (coincident lines). Just look for an equation that is an exact multiple of the other.