Pair of Linear Equations in Two Variables
This domain tests graphical interpretation, consistency conditions, and the substitution, elimination and cross-multiplication methods for two equations in x and y.
Core rules
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
- Unique solution when a₁/a₂ ≠ b₁/b₂; the lines intersect at one point.
- Infinitely many solutions when a₁/a₂ = b₁/b₂ = c₁/c₂; the lines coincide.
- No solution when a₁/a₂ = b₁/b₂ ≠ c₁/c₂; the lines are parallel. Elimination makes one coefficient equal and subtracts; cross-multiplication uses x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁). Graphically, the number of intersection points equals the number of solutions.
Worked example
Solve 3x + 4y = 10 and 2x − 2y = 2. Simplify the second equation: x − y = 1, so x = y + 1. Substitute: 3(y + 1) + 4y = 10 → 7y + 3 = 10 → y = 1. Then x = 2. Check in the first equation: 3(2) + 4(1) = 10. The solution is (2, 1).
Traps
- Comparing ratios without first writing both equations in the same standard form.
- Mixing up the c-ratio condition: equal ratios including c means coincident lines, not parallel lines.
- Sign errors when subtracting equations during elimination.
- Giving only the algebraic solution when a graphical solution is requested.