Linear Equations in Two Variables
What this domain tests
This domain tests pairs of linear equations: graphical and algebraic solutions, consistency conditions, and word problems. AP SSC frequently asks you to decide whether a pair has one solution, no solution or infinitely many.
Core rules
Write both equations as a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.
- Unique solution: a₁/a₂ ≠ b₁/b₂. The lines intersect.
- Infinitely many solutions: a₁/a₂ = b₁/b₂ = c₁/c₂. The lines coincide.
- No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂. The lines are parallel.
- Substitution: express one variable from one equation and substitute into the other.
- Elimination: make the coefficients of one variable equal and opposite, then add.
Worked example
Solve 2x + y = 7 and x − y = 2. Add the equations to eliminate y: 3x = 9, so x = 3. Substitute into x − y = 2: 3 − y = 2, so y = 1. Check in 2x + y: 6 + 1 = 7. The solution is (3,1). The coefficient ratios give 2/1 ≠ 1/(−1), confirming a unique solution.
Traps
- Comparing ratios before writing both equations in standard form.
- Treating parallel lines as having infinitely many solutions.
- Dividing by a coefficient that could be zero without checking.
- Accepting a solution without substituting it into both original equations.
- In word problems, forgetting to define both variables clearly.