Introduction:
In mathematics, a pair of linear equations with two variables is a set of two equations in which each equation is linear, meaning that the variables are raised only to the first power. The problem is to find the values of the variables that make both the equations true simultaneously.

Explanation:
The given pair of linear equations are:
5x + 7y = 10
2x + 3y = p

For a unique solution, the two equations must represent two straight lines that intersect at a single point in the xy-plane. If the two lines are parallel, they will never intersect, and if they coincide, there will be an infinite number of solutions.

Method:
To find the value of p for which the given pair of linear equations has a unique solution, we need to find the condition for the two lines to intersect at a single point.

Solution:
If we solve the given pair of linear equations simultaneously, we get:
x = (3p - 14)/11
y = (10 - 2p)/11

The above equations represent the lines in slope-intercept form. The slope of the first line is -5/7, and the slope of the second line is -2/3.

For the two lines to intersect at a single point, they must have different slopes. Therefore, we need to find the value of p for which the slopes of the lines are different.

-5/7 ≠ -2/3
-15 ≠ -14.

Therefore, the value of p for which the given pair of linear equations has a unique solution is any value other than -15/14.

Conclusion:
Hence, we can conclude that if p ≠ -15/14, the given pair of linear equations has a unique solution.