Chapter Summary

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  a₁²+b₁² ≠ 0, a₂²+b₂² ≠ 0.

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

4. Algebraic Methods : We have discussed the following methods for finding the solution(s) of a pair of linear equations :
(i) Substitution Method
(ii) Elimination Method
(iii) Cross-multiplication Method

5. 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.

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