Linear Equations in two Variables

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A linear equation in one variable is an equation with only one unknown variable that can be written in the form ax + b = 0, where a ≠ 0. Example: 2x + 5 = 0, x + 2 = 0, 3y - 7 = 0. Such equations have

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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 = 12, x + y = 17

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The standard form is ax + by + c = 0, where: - a, b, c are real numbers - a and b are not both zero (at least one of a or b must be non-zero) - x and y are variables

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A linear equation in two variables has infinitely many solutions. This is because for any value chosen for one variable, we can find a corresponding value for the other variable that satisfies the equ

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A solution is an ordered pair (x, y) of values that satisfies the equation when substituted. For example, if (3, 2) is a solution of 2x + 3y = 12, then 2(3) + 3(2) = 6 + 6 = 12, which is true.

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Solution method: Choose a value for x, then find y. Solution 1: Let x = 2, then 2 + y = 5, so y = 3. Solution: (2, 3) Solution 2: Let x = 0, then 0 + y = 5, so y = 5. Solution: (0, 5) We can verify:

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Given: 2x + 3y = 18 and x = 4 Substitute x = 4: 2(4) + 3y = 18 8 + 3y = 18 3y = 18 - 8 3y = 10 y = 10/3 Therefore, the solution is (4, 10/3).

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(i) 2x + 3y = 5 ✓ - This is linear (degree 1 in both variables) (ii) x² + y = 4 ✗ - This is not linear (x² has degree 2) (iii) 3x - 4y + 7 = 0 ✓ - This is linear (can be written as 3x - 4y + 7 = 0) R