Key Points - Pair of Linear Equation in Two Variables

- Algebraic Expression: A combination of constants and variables, connected by four fundamental arithmetical operations of +, −, × and ÷ is called an algebraic expression. For example, 3 x 2 + 4 xy − 5 y 2 is an algebraic expression.

- Equation: An algebraic expression with equal to sign (=) is called the equation. Without an equal to sign, it is an expression only. For example, 3x + 9 = 0 is an equation, but 3x + 9 is an expression.

- Linear Equation: If the greatest exponent of the variable(s) in a equation is one, then equation is said to be a linear equation.

- The most general form of a pair of linear equations is:
 a1x + b1 y + c1 = 0 a2 x + b2 y + c2 = 0 Where a1 , a2 , b1 , b2 , c1 , c2 are real numbers and a1
2 + b1 a12 + b12 ≠ 0, a 22 + b 22 ≠ 0.

- The graph of a pair of linear equations in two variables is represented by two lines;
(i) If the lines intersect at a point, the pair of equations is consistent. The point of intersection gives the unique solution of the equation.
(ii) If the lines coincide, then there are infinitely many solutions. The pair of equations is consistent. Each point on the line will be a solution.
(iii) If the lines are parallel, the pair of the linear equations has no solution. The pair of linear equations is inconsistent.

If a pair of linear equations is given by a1x + b1Y + c1 = 0
 and a 2 x + b 2 y + c2 = 0
 (i)  a1 b1
≠
⇒ the pair of linear equations is consistent. (Unique solution). a 2 b2
 (ii)  a1 b1 c1
=
≠ ⇒ the pair of linear equations is inconsistent (No solution). a 2 b2 c2
 (iii)  a1 b1 c1
=
= ⇒ the pair of linear equations is dependent and consistent (infinitely a 2 b2 c2
 many solutions).