Linear Equations in Two Variables
What the domain tests
This domain tests writing equations in the form ax + by + c = 0, drawing their graphs, solving pairs by substitution or elimination, and deciding whether a system is consistent, inconsistent, or dependent.
Core rules and concepts
- A linear equation in two variables has infinitely many solutions; its graph is a straight line.
- For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
- a₁/a₂ ≠ b₁/b₂ gives one unique solution (intersecting lines).
- a₁/a₂ = b₁/b₂ = c₁/c₂ gives infinitely many solutions (coincident lines).
- a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives no solution (parallel lines).
- Elimination removes one variable by making its coefficients equal and opposite.
Worked example
Solve 2x + 3y = 12 and 4x − y = 10. From the second equation, y = 4x − 10. Substitute into the first: 2x + 3(4x − 10) = 12, so 2x + 12x − 30 = 12 and 14x = 42, giving x = 3. Then y = 4(3) − 10 = 2. The solution is (3, 2). Check in the first equation: 2(3) + 3(2) = 12.
Traps that cost marks
When substituting a negative expression, use brackets: 3(4x − 10), not 3 × 4x − 10. Do not compare ratios before writing both equations with the constant on the same side. For word problems, define the variables clearly and reject negative or fractional answers that do not fit the context. A graph that looks parallel should be confirmed by comparing ratios.