 Hello and welcome to the session. In this session we will discuss the question which says that find the equation of the line through the intersection of x-3y is equal to 1 and 3x-2y plus 1 is equal to 0 and 1 and 2, 2x plus y is equal to 3. Now before starting the solution of this question we should know some results and that are, find the lines a1x plus b1y plus c1 is equal to 0 and a2x plus b2y plus c2 is equal to 0 the line through the intersection of that are these two lines is given by a1x plus b1y plus c1 plus k into a2x plus b2y plus c2 the whole is equal to 0 the line ax plus by plus c is equal to 0 that will be equal to minus the coefficient of x which is a over the coefficient of y which is b so slope will be equal to minus a over b y is equal to n1x plus c and y is equal to n2x plus c are parallel to each other where n1 is the slope of this line and n2 is the slope of this line this means if these lines are parallel to each other then this slope will be equal now this will work out as a key idea for solving all and now we will start with the solution first of all we have to find the equation of the line through the intersection of these two lines so given the equations of the lines 1 is equal to 0 from further written as minus 1 is equal to 0 is 2y plus 1 is equal to 0 the equation of the line through the intersection of the given lines by using this result now let us name this as 1 and this as 2 now the equation of the line section which are given by equation number 1 add to 3y minus 1 plus k into 3x minus 2y plus 1 the whole is equal to 0 which further implies x minus 3y minus 1 plus 3kx minus 2ky plus k is equal to 0 which further implies 3k the whole minus y into 3 plus 2k the whole plus k minus 1 is equal to 0 now let us name this equation as equation number 3 now let the slope of equation number 3 is the slope of any line by using this formula so slope of this line which is given by equation number 3 that is m1 is equal to minus coefficient of x which is 1 plus 3k over 1 coefficient of y which is minus of 3 plus 2k the whole is equal to 1 plus 3k over 1 3 plus 2k equation of the line is also given to us also given the equation of the line plus y is equal to 3 now let us name this as equation number 4 now let n which is given by equation number 4 coefficient of x which is 2 over coefficient of y which is 1 which is further equal to minus 2 now we have designed the equation of the line through the intersection of these two lines and parallel to this line also if two lines are parallel to each other then their slopes are equal now we have the equation of the line through the intersection of the given lines which is given by equation number 3 the equation number 4 parallel to each other that means the slope of the line 3 which is this line line 4 with the lines given by equation number 3 slopes now the slope of the equation number 3 is this which is given by m1 and the slope of equation number 4 m1 is equal to m2 now this implies it will be 1 plus 3k other implies on first multiplying 1 plus 3k is equal to minus 2 into 3 plus 2k the whole 2 plus 3k is equal to is equal to minus 6 minus 1 7k is equal to minus 7 which further gives k is equal to minus 7 by 7 which is equal to minus 1 now let us name equation which is the equation of the line through the intersection of the lines 1 and 2 as we are putting equal to minus 1 in 2 is 3y minus 1 plus of minus 1 into 3h minus 2y plus 1 the whole is equal to 0 other implies x minus 3y minus 1 it will be minus of 3x minus 2y plus 1 the whole is equal to 0 this 1 minus 3x plus 2y minus 1 is equal to 0 which further implies minus 2x minus y minus 2 is equal to 0 and this further gives 2x plus y plus 2 the equal equation of the line and this is the solution of this question that's all for this session hope you all have enjoyed this session