- For any linear equation, each solution (x, y) corresponds to a point on the line. General form is given by ax + by + c = 0.
- The graph of a linear equation is a straight line.
- 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: a
_{1}x + b_{1}y + c_{1}= 0; a_{2}x + b_{2}y + c_{2}= 0

where a_{1}, a_{2}, b_{1}, b_{2}, c_{1}and c_{2}are real numbers, such that a_{1}^{2}+ b_{1}^{2}≠ 0, a_{2}^{2}+ b_{2}^{2}≠ 0. - A pair of values of variables ‘x‘ and ‘y’ which satisfy both the equations in the given system of equations is said to be a solution of the simultaneous pair of linear equations.
- A pair of linear equations in two variables can be represented and solved, by

(i) Graphical method

(ii) Algebraic method

**(i) Graphical method.** The graph of a pair of linear equations in two variables is presented by two lines.

**(ii) Algebraic methods.** Following are the methods for finding the solutions(s) of a pair of linear equations:

- Substitution method
- Elimination method
- Cross-multiplication method.

- There are several situations which can be mathematically represented by two equations that are not linear to start with. But we allow them so that they are reduced to a pair of linear equations.
**Consistent system.**A system of linear equations is said to be consistent if it has at least one solution.**Inconsistent system.**A system of linear equations is said to be inconsistent if it has no solution.

**CONDITIONS**** FOR CONSISTENCY**

Let the two equations be:

a_{1}x + b_{1}y + c_{1} = 0

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

Then,

Relationship between coeff. or the pair of equations | Graph | Number of Solutions | Consistency of System |

Intersecting lines | Unique solution | Consistent | |

Parallel lines | No solution | Inconsistent | |

Co-incident lines | Infinite solutions | Consistent |