#### Mathematics (NTSE/Olympiad)

# Linear Equation In Two Variable

__Solutions of Pair of Lines in Two Variable__

There are three types of solutions of Pair of Lines in Two Variable, these are

- Unique solution
- Infinitely many solutions
- No solution.

**(A) Situation of Consistent:** If a system of simultaneous linear equations has at least one solution then the system is said to be consistent.

**(i) Consistent equations with unique solution :** The graphs of two equations intersect at a unique point. For example. Consider the pair of equation of lines x + 2y = 4 and 7x + 4y = 18.

The graphs of pair of lines of these equations intersect each other at the point (2, 1) i.e., x = 2, y = 1. Hence, the equations are consistent with unique solution.

**(ii) Consistent equations with infinitely many solutions : **The graphs (lines) of the two equations will be coincident. For example. Consider the pair of equations 2x + 4y = 9 ⇒ 3x + 6y = 27/2

The graphs of the above equations coincide. Coordinates of every point on the lines are the solutions of the equations. Hence, the given equations are consistent with infinitely many solutions.

**(B) Inconsistent Equation :** If a system of simultaneous linear equations has no solution, then the system is said to be inconsistent.

**No Solution :** The graph (lines) of the two equations are parallel. For example. Consider the pair of equation of lines are 4x + 2y = 10 and 6x + 3y = 6

The graphs (lines) of the given equations are parallel. They will never meet at a point. So, there is no solution. Hence, the equations are inconsistent.

S. No |
Graph of Two Equations |
Types of Equations |

1. |
Intersecting Lines |
Consistent, With Unique Solution |

2. |
Coincident |
Consistent with number of infinite solutions |

3. |
Parallel Lines |
Inconsistent or No Solution is found |