Pair of Linear Equations: Consistency Conditions and Solution Methods
Pair of Linear Equations in Two Variables is tested through a graphical-or-consistency question, a 3-mark solution by substitution, elimination or cross-multiplication, and a 3–4 mark word problem.
Consistency from coefficients
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
- a₁/a₂ ≠ b₁/b₂: intersecting lines, exactly one solution (consistent).
- a₁/a₂ = b₁/b₂ ≠ c₁/c₂: parallel lines, no solution (inconsistent).
- a₁/a₂ = b₁/b₂ = c₁/c₂: coincident lines, infinitely many solutions (dependent).
Graphically the solution is the point of intersection.
Worked example
Solve 2x + y = 8 and x + 3y = 9 by substitution. From the first, y = 8 − 2x. Substituting: x + 3(8 − 2x) = 9 → x + 24 − 6x = 9 → −5x = −15 → x = 3; then y = 8 − 6 = 2. So x = 3, y = 2 - the lines meet at (3, 2).
Word problems
Set up two variables and two equations, solve, then answer in words. Standard types: sum and difference of two numbers; present ages of two people; a two-digit number written as 10x + y with the digit-reversal condition; boat speed with and against the stream; a fixed charge plus a per-day charge.
Traps that cost marks
- Test all three ratios; two equal ratios alone do not prove infinitely many solutions.
- Comparing only a₁/a₂ with b₁/b₂ tells you nothing when they are equal - c decides.
- Word problems lose the final mark when the answer is not restated in words, or a negative age or speed is not rejected.
- Cross-multiplication requires both equations in the form (…) = 0 first; a sign slip in c flips the answer.