01 · STANDARD FORM
Pair of Linear Equations in Two Variables
The general form for a pair of linear equations in two variables x and y is:
a₁x + b₁y + c₁ = 0a₂x + b₂y + c₂ = 0where a₁, b₁, c₁, a₂, b₂, c₂ are real numbers such that a₁² + b₁² ≠ 0 and a₂² + b₂² ≠ 0.
02 · RATIO CONDITIONS
The 3 Cases of Consistency
We can determine whether a pair of linear equations has a solution without drawing graphs by comparing the ratios a₁/a₂, b₁/b₂, and c₁/c₂:
Case 1
a₁/a₂ ≠ b₁/b₂
Lines intersect at a single point ⟹ Exactly 1 unique solution (Consistent).Case 2
a₁/a₂ = b₁/b₂ = c₁/c₂
Lines are coincident (overlap) ⟹ Infinitely many solutions (Consistent & Dependent).Case 3
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Lines are parallel ⟹ No solution (Inconsistent).03 · SOLVED EXAMPLE
Step-by-Step Problem Solving
EXAMPLE 1
Check consistency of 2x + 3y - 9 = 0 and 4x + 6y - 18 = 0
Step 1: Write down coefficients:
a₁ = 2, b₁ = 3, c₁ = -9
a₂ = 4, b₂ = 6, c₂ = -18
Step 2: Compare ratios:
a₁/a₂ = 2/4 = 1/2
b₁/b₂ = 3/6 = 1/2
c₁/c₂ = -9/-18 = 1/2
b₁/b₂ = 3/6 = 1/2
c₁/c₂ = -9/-18 = 1/2
Since a₁/a₂ = b₁/b₂ = c₁/c₂ = 1/2:
The lines are coincident and the pair of equations has infinitely many solutions.
04 · QUICK SUMMARY
Summary Table
a₁/a₂ ≠ b₁/b₂
Intersecting lines (1 Solution)
a₁/a₂ = b₁/b₂ = c₁/c₂
Coincident lines (Infinite Solutions)
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Parallel lines (No Solution)