Linear Equations in two variables-Notes

An equation which can be put in the form

            ax + by + c = 0

where a, b and c are real numbers {a, b ≠ 0) is called a linear equation in two variables ‘x’ and ‘y’

•    General Form of a Pair of Linear Equations in Two Variables

General form of a linear pair of equations in two variables is:

            a₁x + b₁y + c_x - 0 and

            a₂x + b₂y + c₂ = 0

where a₁ b₁, c₁, a₂, b₂, c₂ are real numbers such that

            a₁² + b₁² 0 and a₂² + b₂² 0

The solution of a linear equation in two variables ‘x’ and ‘y’ is a pair of values (one for ‘x’ and other for ‘y’)

which makes the two sides of the equation equal.

There are two methods to solve a pair of linear equations:

            (i) algebraic method

            (ii) graphical method.

We have already studied (i) Substitution method and (ii) Elimination method. Here, we will study cross-multiplication method also.

            If a₁ x + b₁ y + c₁ = 0

                  a₂ x + b₂ y + c₂ = 0

form a pair of linear equations, then the following three situations can arise:

       (i) If then the system is consistent.

       (ii) If then the system is inconsistent.

       (ii) If then the system is dependent and consistent.

       (i) If the graphs of two equations of a system intersect at a point, the system is said to have a unique solution, i.e., the system is consistent.

       (ii) If the graphs of two equations of a system are two parallel lines, the system is said to have no solution, i.e., the system is inconsistent.

       (iii) When the graphs of two equations of a system are two coincident lines, the system is said to have infinitely many solutions, i.e., the system is consistent and dependent.