Overview
This lecture covers the chapter "Pair of Linear Equations in Two Variables," including definitions, graphical and algebraic solution methods, word problems, and key applications such as age, speed-distance, and digit problems.
Introduction to Linear Equations in Two Variables
- A linear equation in two variables is of the form ax + by = c, where a, b, and c are constants and x, y are variables.
- "Linear" means the degree (highest power) of variables is one.
- The "pair" refers to two such equations involving the same variables (x and y).
Graphical Representation and Types of Solutions
- Each linear equation graphs as a straight line.
- Three cases for two lines:
- Intersecting lines: one unique solution (consistent, independent).
- Coincident lines: infinite solutions (consistent, dependent).
- Parallel lines: no solution (inconsistent).
Conditions for Consistency and Inconsistency
- For equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂:
- If (a₁/a₂) ≠ (b₁/b₂): one solution (intersecting).
- If (a₁/a₂) = (b₁/b₂) = (c₁/c₂): infinite solutions (coincident).
- If (a₁/a₂) = (b₁/b₂) ≠ (c₁/c₂): no solution (parallel).
- Equations should be in standard form with constants on the right.
Solving Pair of Linear Equations: Methods
- Substitution method: Solve one equation for a variable, substitute into the other.
- Elimination method: Add or subtract equations to eliminate a variable, then solve for the other.
- Both methods yield the same solution.
Word Problems and Applications
- Age problems: Translate relationships into linear equations and solve for present ages.
- Digit problems: Represent two-digit numbers as 10x + y and set up equations based on digit relations or reversals.
- Speed-Distance-Time problems: Use distance = speed × time; construct equations for different speed or direction scenarios.
- Area/Perimeter problems: Set up equations using formulas (e.g., perimeter=2(l+b), area=l×b).
Key Terms & Definitions
- Variable — A symbol (like x or y) representing a quantity that can change.
- Degree — The highest power of the variable(s) in the equation (degree 1 for linear).
- Consistent system — At least one solution exists (unique or infinite).
- Inconsistent system — No solution exists.
- Substitution method — Solving by replacing one variable with an expression from another equation.
- Elimination method — Solving by adding/subtracting equations to eliminate a variable.
Action Items / Next Steps
- Practice solving pairs of linear equations using both substitution and elimination.
- Attempt additional word problems on age, digits, and speed-distance.
- Review homework and examples provided in the lecture.
- Join the class Telegram group for updates and supplementary materials.
- Prepare for the next lesson on Quadratic Equations.