Pair of Linear Equations in Two Variables

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A linear equation in two variables is an equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers, and a and b are not both zero. Examples: 2x + 3y - 6 = 0, x - y +

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A pair of linear equations in two variables consists of two linear equations involving the same two variables. Example: 2x + 3y = 6 and x - y = 1. The solution is the common values of x and y that sat

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A pair of linear equations that has at least one solution is called a consistent pair. This includes cases with exactly one solution (unique) or infinitely many solutions. Example: x + y = 3 and 2x -

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A pair of linear equations that has no solution is called an inconsistent pair. This happens when the lines representing the equations are parallel. Example: x + y = 3 and x + y = 5 (parallel lines, n

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A dependent pair of linear equations consists of two equations that are equivalent and represent the same line. Such pairs have infinitely many solutions. Example: 2x + 4y = 8 and x + 2y = 4 are depen

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1. Lines intersect at one point → Unique solution (consistent) 2. Lines are parallel → No solution (inconsistent) 3. Lines coincide (same line) → Infinitely many solutions (dependent/consistent)

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When a₁/a₂ ≠ b₁/b₂, the lines intersect at exactly one point, giving a unique solution. The pair is consistent. Example: x + y = 3 and 2x - y = 0 have different slopes, so they intersect once.

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When a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel with no intersection point. The pair is inconsistent. Example: 2x + 3y = 6 and 4x + 6y = 18 (same slope, different y-intercepts).