Pair of Linear Equations — Quick Revision
A pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 can be solved graphically or algebraically. Graphically, the two lines may intersect at one point (a unique solution — consistent), be parallel (no solution — inconsistent) or coincide (infinitely many solutions — dependent). The ratios of the coefficients tell us which case applies without drawing: if a₁/a₂ ≠ b₁/b₂ there is a unique solution; if a₁/a₂ = b₁/b₂ ≠ c₁/c₂ there is no solution; if all three ratios are equal there are infinitely many solutions. Substitution and elimination are the two standard algebraic methods.
Key Points
- •a₁/a₂ ≠ b₁/b₂ ⇒ unique solution (intersecting, consistent)
- •a₁/a₂ = b₁/b₂ ≠ c₁/c₂ ⇒ no solution (parallel, inconsistent)
- •a₁/a₂ = b₁/b₂ = c₁/c₂ ⇒ infinitely many (coincident, dependent)
- •Algebraic methods: substitution, elimination
- •Graph of each equation is a straight line