Pair of Linear Equations in Two Variables
Gujarat Board · Class 10 · Mathematics
Quick revision notes for Pair of Linear Equations in Two Variables — Gujarat Board Class 10 Mathematics. Key concepts, formulas, and definitions for last-minute revision.
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Introduction and Basic Concepts
- A linear equation in two variables x and y has the general form ax + by + c = 0, where a, b, c are real numbers and a² + b² ≠ 0
- A pair of linear equations consists of two such equations that need to be solved simultaneously
- Real-life situations often lead to pairs of linear equations, like cost problems, age problems, speed-time problems
Graphical Method of Solution
- Each linear equation represents a straight line when plotted on a coordinate plane
- The solution of the pair is the point of intersection of the two lines (if they intersect)
- Three possible cases: lines intersect at one point, lines are parallel, or lines coincide
Nature of Solutions Based on Coefficients
- For equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
- If a₁/a₂ ≠ b₁/b₂, then unique solution (intersecting lines)
- If a₁/a₂ = b₁/b₂ = c₁/c₂, then infinitely many solutions (coincident lines)
Substitution Method
- Express one variable in terms of the other from one equation
- Substitute this expression into the second equation
- Solve the resulting equation in one variable
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