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# Find the real numbers x and y if $(x - iy)(3 + 5i)$ is the conjugate of $- 6 - 24i$

Last updated date: 14th Sep 2024
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Hint: According to the question given in the question we have to find the real numbers x and y if $(x - iy)(3 + 5i)$ is the conjugate of $- 6 - 24i$. So, first of all we have to determine the multiplication of the terms of the expression $(x - iy)(3 + 5i)$.
Now, we have to separate the real and imaginary terms of the expression obtained and as given that the conjugate of $(x - iy)(3 + 5i)$ is $- 6 - 24i$ so we have to obtained the inverse of the $- 6 - 24i$ by which we can obtained the values x and y.

Formula used: $\Rightarrow {i^2} = - 1.......................(A)$

Complete step-by-step solution:
Step 1: First of all we have to multiply the terms of the expression $(x - iy)(3 + 5i)$ as mentioned in the solution hint. Hence,
$\ = (x - iy)(3 + 5i) \\ = 3x + 5xi - 3yi - 5y{i^2}...............(1)$
Step 2: Now, to solve the expression we have to use the identity (1) as mentioned in the solution hint.
$\ = 3x + 5xi - 3yi - 5y( - 1) \\ = 3x + 5xi - 3yi + 5y \\ = (3x + 5y) + i(5x - 3y)...........(2)$
Step 3: Now, as we know that the conjugate of $(x - iy)(3 + 5i)$ is $- 6 - 24i$ hence, we have to find the inverse of $- 6 - 24i$ which is $\overline { - 6 - 24i} = - 6 + 24i$ so, on substituting the inverse obtained in the expression (2) as obtained in the solution step 2.
$= (3x + 5y) + i(5x - 3y) = - 6 + 24i$……………..(3)
Step 4: Now, to obtain the values of x and y we have to compare the real and imaginary terms of the expression (3) as obtained in the solution step 3. Hence,
$\Rightarrow (3x + 5y) = - 6............(4)$
$\Rightarrow (5x - 3y) = 24.................(5)$
Step 5: Now, we have to solve the obtained expressions (4) and (5) but before that we have to multiply the equation (4) with 5 and equation (5) with 3. Hence, obtained equations are,
$\ \Rightarrow 5(3x + 5y) = 5 \times ( - 6) \\ \Rightarrow 15x + 25y = - 30............(6)$
And,
$\ \Rightarrow 3(5x - 3y) = 3 \times 24 \\ \Rightarrow 15x - 9y = 72.................(7)$
Step 6: Now, we have to subtract the expression (7) from the expression (6). Hence,
$\ \Rightarrow 15x + 25y - 15x + 9y = - 30 - 72 \\ \Rightarrow 34y = - 102 \\ \Rightarrow y = - \dfrac{{102}}{{34}} \\ \Rightarrow y = - 3$
Step 7: Now, to obtain the value of x we have to substitute the value of y in equation (4). Hence,
$\ \Rightarrow 3x + 5( - 3) = - 6 \\ \Rightarrow 3x = - 6 + 15 \\ \Rightarrow x = \dfrac{9}{3} \\ \Rightarrow x = 3$

Hence, with the help of identity (A) as mentioned in the solution hint we have obtained the values of x and y for if $(x - iy)(3 + 5i)$ is the conjugate of $- 6 - 24i$ are $x = 3$ and $y = - 3$

Note: If the conjugate for any complex equation/expression is given then to solve the given equation/expression it is necessary to find the inverse of that given conjugate as if the conjugate of the given equation/expression is x then the inverse of that given conjugate is $\overline x$
If the imaginary term i is multiplied with i or on squaring i means ${(i)^2}$ we will obtain the real term as -1.