Linear Equations in Two Variables

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A linear equation in one variable is an equation where the degree (highest power) of the variable is 1. Examples: m + 3 = 5, 3y + 8 = 22, x/3 = 2. These equations have only one solution.

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A linear equation in two variables is an equation with two variables where the degree of both variables is 1. General form: ax + by + c = 0, where a, b, c are real numbers and a and b cannot both be z

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A linear equation in two variables has infinitely many solutions. Example: x + y = 14 has solutions like (9,5), (7,7), (8,6), (4,10), (-1,15), (2.6,11.4), etc. Each solution is written as an ordered p

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Simultaneous equations are two or more linear equations in two variables that are considered at the same time. We need to find the common solution that satisfies all equations simultaneously. Example:

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Adding the equations: (x + y) + (x - y) = 14 + 2 → 2x = 16 → x = 8. Substituting x = 8 in first equation: 8 + y = 14 → y = 6. Solution: (8, 6)

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The elimination method involves eliminating one variable by adding or subtracting the equations (after making coefficients equal if needed). This gives an equation in one variable, which can be solved

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Multiply second equation by 3: 9x - 3y = 3. Add with first equation: (2x + 3y) + (9x - 3y) = 7 + 3 → 11x = 10 → x = 10/11. Substitute: 2(10/11) + 3y = 7 → 3y = 7 - 20/11 = 57/11 → y = 19/11. Solution:

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The substitution method involves expressing one variable in terms of the other from one equation, then substituting this expression into the other equation. This eliminates one variable, creating an e