 Hello and welcome to the session. In this session we will discuss a question which says that for what value of A, the free straight lines A x plus 2 y minus 5 is equal to 0 to x minus 3 y plus 1 is equal to 0, x plus y minus 2 is equal to 0 are congruent. Now before starting the solution of this question we should know our result and that is the three drawn lines are congruent they have a common point of intersection. Now this result will work out as a key idea for solving out this question. And now we will start with the solution. Now in the question it is given that these three lines are congruent and we have to find the value of A. Now for finding out the value of A first of all we will solve these two equations for finding out the point of intersection. Then we will get the coordinates of the point of intersection in the equation A x plus 2 y minus 5 is equal to 0. Now as these three lines are congruent this means that the point of intersection that is the coordinates of point of intersection will satisfy this equation also. This we will find out the value of A. Now given the equations of the lines are 2 x minus 3 y plus 1 is equal to 0 and x plus y minus 2 is equal to 0 which can be further written as 2 x minus 3 y is equal to minus 1 plus y is equal to 2. Now let this be equation number 1 and this be equation number 2. So multiplying the equation number 2 by 2 minus 3 y is equal to minus 1 and the second equation will become 2 x plus 2 y is equal to 4. Now let this be equation number 3 and this be equation number 4. Now subtracting 3 from 4 we get y the whole minus 2 x minus 3 y the whole is equal to 4 minus of minus 1 will become 4 plus 1. Now this further implies on solving 5 y is equal to 5 which further gives y is equal to 1. Now this is the equation number 2. So putting y is equal to 1 in equation number 2 we get x plus 1 minus 2 is equal to 0 which implies x is equal to 1. So we have got y is equal to 1 and x is equal to 1 therefore the point of intersection that is of the lines which are given by equation number 1 and 2 is 1 1. Now we know that the 3 lines are concurrent if they have a common point of intersection that the third line which is given by the equation a x plus 2 y minus 5 is equal to 0 will also pass through the point of intersection these 2 lines. The third line which is given by this 2 y minus 5 is equal to 0 also passes that is to the point 1 1 that means this point will satisfy the equation a x plus 2 y minus 5 is equal to 0 therefore now putting x is equal to 1 and y is equal to 1 in this equation it will be a into 1 plus 2 into 1 minus 5 is equal to 0 which implies a plus 2 minus 5 is equal to 0 which implies a is equal to the value of a that is for this value of a the given 3 lines are concurrent. So this is the solution of the given equation and that is all for this session hope you all have enjoyed the session.