Formulas of pair of linear equations in two variables

Answer:

(1) An equation in the form ax + by + c = 0, where a, b and c are real numbers, and a≠ 0 and b ≠ 0, is called a linear equation in two variables x and y.

For Example: 2x + 3y + 7 = 0, where a = 2, b = 3, c =5 are real numbers. So, given equation is a linear equation in two variables.

(2) Each solution (x, y) of a linear equation in two variables, ax + by + c = 0, corresponds to a point on the line representing the equation, and vice versa.

For Example: 2x + 3y = 5 has (1, 1) as its solution. So,(1, 1) will lie on the line 2x + 3y = 5.

(3) The general form for a pair of linear equations in two variables x and y is a1x + b1 y + c1 = 0 and a2x + b2 y + c2 = 0, where a1 , b1 , c1 , a2 , b2 , c2 are all real numbers and a12 + b12 ≠ 0, a22 + b22 ≠ 0.

For Example: 2x + 3y – 7 = 0 and 9x – 2y + 8 = 0 forms a pair of linear equations.

(4) A pair of linear equations which has no solution is called an inconsistent pair of linear equations. In this case, the lines may be parallel a1/a2 = b1/b2 ≠ c1/c2.

For Example: x + 2y – 4 = 0 and 2x + 4y – 12 = 0 are parallel lines.

(5) A pair of linear equations in two variables, which has a solution, is called a consistent pair of linear equations. In this case, the lines may intersect in a single point and a1/a2 ≠ b1/b2.

For Example: x – 2y = 0 and 3x + 4y – 20 intersects each other at unique point (4, 2).

(6) A pair of linear equations which are equivalent has infinitely many distinct common solutions. Such a pair is called a dependent pair of linear equations in two variables. In this case, the lines may be coincident and a1/a2 = b1/b2 = c1/c2.

For Example: 2x + 3y – 9 = 0 and 4x + 6y – 18 = 0 are coincident lines.

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