 Hello and welcome to the session. In this session we will discuss a question which says that find the real numbers x and y if x plus y iota the whole into 7 plus 6 iota the whole is the constant of minus 21 plus 13 iota. Now before starting the solution of this question we should know a result and that is if z is equal to a plus b iota the complex number then conjugate of z is equal to a minus b iota and it can also be written as the conjugate of z is equal to a minus b iota. This means that the conjugate of a complex number is formed by changing the sign of the imaginary part. Now this result will work out as a key idea for solving out this question and now we will start with the solution. Now we have to find the real numbers x and y if this condition is given to us. Now using this result which is given in the key idea the conjugate of minus 21 plus 13 iota minus 21 minus 13 iota that is by changing the sign of the imaginary part we are getting the conjugate of a number. Now according to the given condition y iota the whole into 7 plus 6 iota the whole is the conjugate of this number that means this is equal to minus 21 minus 13 iota. Further on solving this implies x iota plus 7 y iota plus 6 y iota square is equal to minus 21 minus 13 iota. Now this implies 7 x plus x iota plus 7 y iota plus 6 y into now iota square is equal to minus 1 so it would be minus 1 here is equal to minus 21 minus 13 iota. Now this implies 7 x plus 6 x iota plus 7 y iota minus 6 y is equal to minus 21 minus 13 iota. Now this can also be written as 7 x minus 6 y the whole iota is equal to minus 13 iota. Now we know that complex numbers b iota where a b are real numbers and real part and b is called the imaginary part. Now here in this case equating that are these we have 7 x minus 6 y is equal to minus 21 the imaginary part we have is equal to minus 13 i is equal to minus 21 7 y is equal to minus 13. Now let us name this as equation number one and this has equation number two these two equations simultaneously and for this we will multiply equation number one equation number two by 6 49 x minus 42 minus 147 equation by 6 we get plus 42 y is equal to minus 78. Now let us name this as equation number three and this as equation number four now adding equation number three and four x minus 42 y plus 36 x plus 42 y is equal to minus 147 plus half minus 78. Now these terms are cancelled with each other and this implies 36 x is equal to 85 x is equal to minus 147 plus half minus will be minus so it will be minus 147 minus 78 which is equal to minus 225 further this implies 225 over 85. Now 5 into 17 is 85 and 5 into 45 so this implies x is equal to minus 17 now this is my equation number one now putting 55 by 17 in equation number one we get minus 45 by 17 minus 6 y is equal to minus 21 now this implies 17 minus 6 y is equal to minus 21 on taking the same this implies minus 350 minus 102 y over 17 is equal to minus 21 now on cross multiplying this implies minus 315 minus 102 y is equal to 17 into minus 21 which is equal to minus 357 and this implies minus 102 y is equal to minus plus 350 which further means minus 102 y is equal to minus 42 minus 102 now here these are cancelled with each other and here is 42 and 2 into 51 is 102 further is 21 and 3 into 17 is 51 so this implies y is equal to 7 by 17 this is the value of y we get x is equal to minus 45 by 17 to 7 by 17 the solution of the given question and that's all for the session hope you all have enjoyed the session