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 - 5 = 0, x - 4y +…

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A pair of linear equations in two variables consists of two linear equations with the same two variables. For example: a₁x + b₁y + c₁ = 0 a₂x + b₂y + c₂ = 0 Where x and y are the same variables in bot…

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A pair of linear equations in two variables that has at least one solution is called a consistent pair of linear equations. This means the equations can be satisfied by some values of x and y.

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A pair of linear equations in two variables that has no solution is called an inconsistent pair of linear equations. This occurs when the lines represented by the equations are parallel and never inte…

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A pair of linear equations that are equivalent and have infinitely many solutions is called a dependent pair of linear equations. The equations represent the same line (coincident lines). A dependent …

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For intersecting lines (unique solution): a₁/a₂ ≠ b₁/b₂ This means the ratios of coefficients of x and y are not equal, ensuring the lines intersect at exactly one point.

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For parallel lines (no solution): a₁/a₂ = b₁/b₂ ≠ c₁/c₂ The ratios of coefficients of x and y are equal, but the ratio of constant terms is different, making the lines parallel.

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For coincident lines (infinitely many solutions): a₁/a₂ = b₁/b₂ = c₁/c₂ All ratios of corresponding coefficients are equal, meaning the equations represent the same line.