Learning Objectives
- Represent a pair of linear equations graphically and algebraically
- Determine consistency of a system of linear equations
- Solve pairs of linear equations by substitution, elimination, and cross-multiplication methods
- Solve word problems using systems of linear equations
Key Concepts
General Form
A pair of linear equations in two variables x and y:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Graphical Representation and Consistency
- Intersecting lines (unique solution / consistent): a₁/a₂ ≠ b₁/b₂
- Coincident lines (infinitely many solutions / dependent): a₁/a₂ = b₁/b₂ = c₁/c₂
- Parallel lines (no solution / inconsistent): a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Algebraic Methods of Solving
Substitution Method: Express one variable from one equation, substitute into the other, and solve.
Elimination Method: Multiply equations by suitable numbers to make the coefficients of one variable equal (or opposite), then add or subtract the equations to eliminate that variable.
Cross-Multiplication Method: For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
x / (b₁c₂ - b₂c₁) = y / (c₁a₂ - c₂a₁) = 1 / (a₁b₂ - a₂b₁)
Summary
A pair of linear equations in two variables can have a unique solution, infinitely many solutions, or no solution. The ratio of coefficients determines the type. Graphically, these correspond to intersecting, coincident, or parallel lines. Substitution, elimination, and cross-multiplication are algebraic methods to find solutions.
Important Terms
- Consistent System
- A system that has at least one solution (intersecting or coincident lines)
- Inconsistent System
- A system that has no solution (parallel lines)
- Dependent System
- A system with infinitely many solutions (coincident lines)
Quick Revision
- a₁/a₂ ≠ b₁/b₂ → unique solution (intersecting lines)
- a₁/a₂ = b₁/b₂ = c₁/c₂ → infinitely many solutions (coincident lines)
- a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → no solution (parallel lines)
- Substitution is easiest when one coefficient is 1
- Elimination is useful when coefficients can be easily matched