 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says find the real numbers x and y if x-iota y into 3 plus 5 iota is the conjugate of minus x-24 iota. For any complex number z is equal to x plus iota y. It is a conjugate which is denoted by z bar is equal to x-iota y. So this is the key idea we are going to use in this problem to solve it. Let us now begin with the solution and we are given that x-iota y into 3 plus 5 iota is a conjugate of minus x-24 iota and conjugate of minus x-24 iota is equal to minus 6 plus 24 iota. So this implies x-iota y into 3 plus 5 iota is equal to minus 6 plus 24 iota. Now let us solve the left hand side. This implies 3x plus iota times 5x minus iota times 3y minus 5y iota square is equal to minus 6 plus 24 i or this implies 3x and here iota square is minus 1 so I am multiplying minus 1 with 5y we get minus 5y and minus into minus is plus so we have plus 5y and now taking i common from these 2 terms we have i into 5x minus 3y it is further equal to minus 6 plus 24 iota. Now comparing the real and imaginary parts we have 3x plus 5y is equal to minus 6 and 5x minus 3y is equal to 24 this we have gone comparing the real and imaginary parts and let this be equation number 1 and this be equation number 2. Now to get the values of x and y we will solve these equations by admitting the coefficients so multiplying equation 1 by 5 and equation 2 by 3 these equations can further be written as first let us write the first equation which can be written as 15x plus 25y is equal to minus 30 and second equation can be written as 15x minus 9y is equal to 72. Now subtracting these 2 terms these 2 cancels out here we have plus and here we have minus and we are subtracting minus 9 from 25 we get here plus sign so on adding we have 34y is equal to now we will add minus 30 with minus 72 and we get minus 102 plus y is equal to minus 102 upon 34 this implies y is equal to minus 3. Now in substituting y is equal to minus 3 in equation 1 we get 3x plus 5 into minus 3 is equal to minus 6 or 3x minus 15 is equal to minus 6 which further implies that 3x is equal to 15 minus 6 or we can say 3x is equal to 9 which further implies that x is equal to 9 upon 3 which is equal to 3 and last we have x is equal to 3 and hence our answers are x is equal to 3 and y is equal to minus 3 so this completes the solution hope you enjoyed it take care and have a good day