Linear Equations

Introduction

If two linear equations have the two same variables, they are called a pair of linear equations in two variables. Following is the most general form of linear equations:

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

Here, a₁, a₂, b₁, b₂, c₁ and c₂ are real numbers such that;

`a_1^2+b_1^2≠0`

`a_2^2+b_2^2≠0`

A pair of linear equations can be represented and solved by the following methods:

Graphical method
Algebraic method

Graphical Method:

For a given pair of linear equations in two variables, the graph is represented by two lines.

If the lines intersect at a point, that point gives the unique solution for the two equations. If there is a unique solution of the given pair of equations, the equations are called consistent.
If the lines coincide, there are indefinitely many solutions for the pair of linear equations. In this case, each point on the line is a solution. If there are infinitely many solutions of the given pair of linear equations, the equations are called dependent (consistent).
If the lines are parallel, there is no solution for the pair of linear equations. If there is no solution of the given pair of linear equations, the equations are called inconsistent.

Algebraic Method:

There are following methods for finding the solutions of the pair of linear equations:

Substitution method
Elimination method
Cross-multiplication method

If a pair of linear equations is given by a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 then following situations can arise.

Situation 1:

`(a_1)/(a_2)≠(b_1)/(b_2)`

In this case, the pair of linear equations is consistent. This means there is unique solution for the given pair of linear equations. The graph of the linear equations would be two intersecting lines.

Situation 2:

`(a_1)/(a_2)=(b_1)/(b_2)≠(c_1)/(c_2)`

In this case, the pair of linear equations is inconsistent. This means there is no solution for the given pair of linear equations. The graph of linear equations will be two parallel lines.

Situation 3:

`(a_1)/(a_2)=(b_1)/(b_2)=(c_1)/(c_2)`

In this case, the pair of linear equations is dependent and consistent. This means there are infinitely many solutions for the given pair of linear equations. The graph of linear equations will be coincident lines.