Maths Class 10 Chapter 3: Pair of Linear Equations in Two Variables

Important Concepts and Formulas

1.      Two linear equations in the same two variables are called a pair of linear equations in two variables. The most general form of a pair of linear equations is

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

where a_1, a_2, b_1, b_2, c_1, c₂ are real numbers such that a₁² + b₁² ≠ 0 and a₂² + b₂² ≠ 0.

2.      A pair of linear equations in two variables can be represented and solved by the following methods:

                                i.            Graphical method

                              ii.            Algebraic method

3.      Graphical method: The graph of a pair of linear equations in two variables is represented by two lines.

                                i.            If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent.

                              ii.            If the lines coincide, then there are infinitely many solutions, i.e., each point on the line being a solution. In this case, the pair the equations is dependent (consistent).

                            iii.            If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is inconsistent.

4.      Algebraic method: We use the following methods to find the solution of a pair of linear equations:

                                i.            Substitutionmethod

                              ii.            Eliminationmethod

                            iii.            Cross-multiplication

To study the above methods in detail -------- Click Here!

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

                                 i.            If a₁/ a₂ ≠ b₁/ b₂, then both the equations have a unique solution and they are consistent.

                               ii.            If a₁/ a₂ = b₁/ b₂ = c₁/c₂, then both the equations have infinitely many solutions and they are dependent and consistent.

                              iii.            If a₁/ a₂ = b₁/ b₂ ≠ c₁/c₂, then both the equations have no solution and the equations are said to be inconsistent.