Pair of Linear Equations in Two Variables

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A linear equation in two variables is an equation of the form ax + by + c = 0, where a, b, c are real numbers and at least one of a or b is not zero (a² + b² ≠ 0). Here x and y are the two variables.

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A pair of linear equations in two variables consists of two linear equations in the same two variables. General form: a₁x + b₁y + c₁ = 0 (a₁² + b₁² ≠ 0) a₂x + b₂y + c₂ = 0 (a₂² + b₂² ≠ 0) Example: 3x

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A solution of a pair of linear equations is a pair of values (x, y) that satisfies both equations simultaneously. For example, if (2, 3) is a solution, then substituting x = 2 and y = 3 in both equati

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1. Unique solution (one solution) - lines intersect at exactly one point 2. Infinitely many solutions - lines are coincident (same line) 3. No solution - lines are parallel and never intersect

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A pair of linear equations has a unique solution when the two lines intersect at exactly one point. This happens when a₁/a₂ ≠ b₁/b₂. Such equations are called consistent and independent equations.

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A pair of linear equations has infinitely many solutions when the lines are coincident (same line). This happens when a₁/a₂ = b₁/b₂ = c₁/c₂. Such equations are called consistent and dependent equation

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A pair of linear equations has no solution when the lines are parallel and never intersect. This happens when a₁/a₂ = b₁/b₂ ≠ c₁/c₂. Such equations are called inconsistent equations.

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For equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0: • Unique solution: a₁/a₂ ≠ b₁/b₂ • Infinitely many solutions: a₁/a₂ = b₁/b₂ = c₁/c₂ • No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂