Linear Equations in Two Variables
This domain tests solving pairs of linear equations graphically and algebraically, deciding consistency from coefficients, and translating word problems into two-variable equations.
Core rules
- Standard form: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0; each graph is a straight line.
- Unique solution when a₁/a₂ ≠ b₁/b₂ (intersecting lines, consistent).
- Infinitely many solutions when a₁/a₂ = b₁/b₂ = c₁/c₂ (coincident lines, dependent).
- No solution when a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel lines, inconsistent).
- Substitution, elimination and cross-multiplication all give the same solution; choose the one that avoids fractions.
Worked example
Solve 2x + y = 8 and x − y = 1. From the second, y = x − 1. Substitute: 2x + (x − 1) = 8, so 3x = 9 and x = 3. Then y = 3 − 1 = 2. Check in the first equation: 6 + 2 = 8. The lines have different slopes, confirming one solution.
Traps
Not testing the ratio condition before solving, so parallel lines are reported as solved; sign errors when transposing c in the standard form; adding equations without matching coefficients; leaving word-problem variables undefined; accepting a negative value for a length, age or speed without comment.