A pair of linear equation which has a common point is known as unique solution.This unique solution is consistent.

Introduction
A pair of linear equations in two variables is a set of two equations that involve two variables, typically represented as x and y. These equations can be written in the form of ax + by = c, where a, b, and c are constants. When these equations have a common point of intersection, they are said to have a unique solution.

Explanation
When solving a pair of linear equations, we aim to find the values of x and y that satisfy both equations simultaneously. If the equations have a common point of intersection, it means there is a unique solution that satisfies both equations. This solution represents the coordinates of the point where the two lines intersect.

Graphical Interpretation
A pair of linear equations can be graphically represented as two straight lines on a coordinate plane. The point of intersection of these lines represents the solution to the system of equations. If the lines intersect at one point, it implies that there is only one solution.

Algebraic Interpretation
Mathematically, the unique solution can be obtained by solving the system of equations using various methods such as substitution, elimination, or matrix methods. These methods involve manipulating the equations to eliminate one variable and subsequently solve for the other variable. The resulting values of x and y represent the unique solution.

Properties of a Unique Solution
1. The unique solution satisfies both equations simultaneously.
2. The solution is a single point in the coordinate plane.
3. The lines represented by the equations are not parallel, as parallel lines do not intersect.

Example
Let's consider the following pair of linear equations:
Equation 1: 2x + 3y = 7
Equation 2: 4x - y = 5

By solving these equations, we can find the unique solution. Using the substitution method, we can solve Equation 2 for y and substitute it into Equation 1:
4x - y = 5
y = 4x - 5

Substituting this value of y into Equation 1:
2x + 3(4x - 5) = 7
2x + 12x - 15 = 7
14x - 15 = 7
14x = 22
x = 22/14
x = 11/7

Substituting the value of x back into Equation 1:
2(11/7) + 3y = 7
22/7 + 3y = 7
3y = 35/7 - 22/7
3y = 13/7
y = 13/21

Therefore, the unique solution to the given pair of linear equations is x = 11/7 and y = 13/21. This solution represents the point of intersection of the two lines.

Conclusion
When a pair of linear equations in two variables has a common point of intersection, it has a unique solution. This unique solution can be obtained by graphically interpreting the point of intersection or algebraically solving the system of equations. The unique solution satisfies both equations simultaneously and represents a single point in the coordinate plane.