Notes for Pair of Linear Equations in Two Variables

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 linear pair of equations in two variables is:

a_{1}x + b_{1}y + c_{x} - 0 and

a_{2}x + b_{2}y + c_{2} = 0

where a_{1} b_{1}, c_{1}, a_{2}, b_{2}, c_{2} are real numbers such that

a_{1}^{2} + b_{1}^{2} 0 and a_{2}^{2} + b_{2}^{2} 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 a_{1} x + b_{1} y + c_{1} = 0

a_{2} x + b_{2} y + c_{2} = 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.**