00:00So, the question here is x and y are real numbers such that 2x divided by 1 plus 3i minus
00:07y divided by 1 minus 2i.
00:10Here, we can say iota is equal to under root minus 1 multiplied by 50 is equal to 31 plus
00:1817 iota.
00:19And we are asked in the question what is the value of 10 into x minus 3y.
00:24So, first of all, we will take calcium.
00:27So, if we take calcium, then we will multiply both of these terms in the denominator.
00:33This left side will be multiplied by right side, right will be multiplied by left.
00:37We all know how to take calcium.
00:39So, 2x multiplied by 1 minus 2i.
00:43So, 2x into 1, 2x into minus 2i is equal to minus 4xi.
00:49minus y into 1y minus y into 3i minus 3y iota divided by 1 plus 3i into 1 minus 2i
01:02is equal to 31 plus 17 iota.
01:05Now, we multiply the denominator term.
01:091 into 1, 1.
01:103i into 1, 3i minus 2i into 1, minus 2i.
01:153i minus 2i is equal to single iota.
01:193i is equal to minus 2i is equal to minus 6i is equal to minus 1, minus 1 into minus
01:286 is equal to 6.
01:316 plus this one from earlier is equal to 7.
01:347 plus 3i minus 2i was iota is equal to 7 plus iota.
01:40So, we arrive at here.
01:43Now, we will solve further.
01:45But, let us solve the right side.
01:48Take this denominator to this side.
01:51So, this will multiply.
01:537 plus 3i into 31 plus 17 iota.
01:577 into 31 is equal to 217.
02:01And 7 into 17 is equal to 119 iota.
02:0617 iota is also here.
02:08Plus iota into 31, 31 iota.
02:12Plus iota into 17 iota.
02:1417 iota is equal to minus 1.
02:18Which comes to minus 17.
02:21Now, 217 minus 17.
02:24219 plus 31 iota is equal to 150 iota.
02:29We can take 50 common from both.
02:32If we take 50 common, it will become 4 plus 3 iota.
02:36We can cancel 50 on left side from this earlier 50 on the right side.
02:42So, it becomes 50, cancel by 50.
02:45So, we get 2x minus 4x iota minus y minus 3y iota is equal to 4 plus 3 iota.
02:52Now, we know in complex number, we can find value by comparing both sides.
02:57So, on the left side, we have the imaginary term minus 4x minus 3y is equal to 3.
03:07First compare iota times, then normal ones.
03:10And normal one is 2x minus y is equal to 4.
03:15So, we have written both equations.
03:172 equations, 2 variables we can solve.
03:202x minus y is equal to 4.
03:222x is equal to 4 plus y.
03:25Multiply it by what?
03:28Minus 2.
03:29Multiply both sides by minus 2.
03:32This becomes minus 4x is equal to minus 8 minus 2y.
03:38Now, put minus 8 minus 2y in value of place of minus 4x.
03:43So, it becomes minus 8 minus 2y minus 3y is equal to 3.
03:50Minus 8 minus 5y is equal to 3.
03:54On further solving, it becomes 5y is equal to 11.
03:59y is equal to minus 5y is equal to 11.
04:04Which means y is equal to minus 11 upon 5.
04:09Then, we solve further.
04:10Put the value of y in equation, which is here.
04:14Minus 11 upon 5 here becomes plus 11 upon 5.
04:19And, if we take it to the right side, it becomes 4 plus 11 upon 5 is equal to 2x.
04:29There was a confusion.
04:31y value is minus 11 upon 5.
04:34Here, it becomes plus 11 upon 5.
04:36If we take it to the right side, it will become 4 minus 11 upon 5.
04:41If we will take calcium, 4 minus 11 upon 5.
04:48Denominator 5 and numerator, this is 4.
04:53So, this becomes, let me do here.
04:582x is equal to 4 minus 11 upon 5.
05:0520 minus 11 upon 5.
05:084 multiplied by 5 is equal to 20.
05:10Is equal to 2x is equal to 9 by 5.
05:14x value, we get 9 by 10.
05:19Now, we have been added 10 into x minus 3y.
05:23If we do 10 into x value, what do we have x value?
05:28x value is here.
05:32We find here only where.
05:34Here, x value is 9 upon 10.
05:36Which means, 10 multiplied by 9 upon 10 minus 3y.
05:42y value is minus 11 upon 5.
05:45Minus 11 into minus 3 is equal to plus 33 upon 5.
05:509 by 10 plus 33 upon 5.
05:52If we take calcium, it becomes, this becomes 66 and this remains 9.
05:5866 plus 9 is equal to 75 upon 10.
06:0275 upon 10 multiplied by 10 is equal to 75.
06:06So, finally, we have value of 10 into x minus 3y is equal to 75.
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