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Summary

In this chapter, we have studied the following points:

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 a1, a2, b1, b2, c1, c2 are real numbers, such that +≠ 0, +≠ 0.

 

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

(i) graphical method

(ii) algebraic method


(i) 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 — each point on the line being a solution. In this case, the pair of 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.


(ii) Algebraic Methods
We have discussed the following methods for finding the solutions of a pair of linear equations :

(i) Substitution Method

(ii) Elimination Method

(iii) Cross-multiplication Method

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) : In this case, the pair of linear equations is consistent.

(ii) : In this case, the pair of linear equations is inconsistent.

(iii) : In this case, the pair of linear equations is dependent and consistent.

6. There are several situations which can be mathematically represented by two equations

that are not linear to start with. But we alter them so that they are reduced to a pair of

linear equations.

Condition

Nature of Equation

Type of Lines

Type of Solution

        Consistent

    Intersecting lines

     Unique solution

        Consistent

    Coincident or         overlapping lines

     Infinitely many    solutions

       Inconsistent

      Parallel lines

        No solution





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