Question

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

Answer 1

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,
$$\begin{gathered} =(x-iy)(3+5i) \\ =3x+5xi-3yi-5y{i}^2...............(1) \end{gathered}$$
Step 2: Now, to solve the expression we have to use the identity (1) as mentioned in the solution hint.
$$\begin{gathered} =3x+5xi-3yi-5y(-1) \\ =3x+5xi-3yi+5y \\ =(3x+5y)+i(5x-3y)...........(2) \end{gathered}$$
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,
$$\begin{gathered} \Rightarrow 5(3x+5y)=5\times (-6) \\ \Rightarrow 15x+25y=-30............(6) \end{gathered}$$
And,
$$\begin{gathered} \Rightarrow 3(5x-3y)=3\times 24 \\ \Rightarrow 15x-9y=72.................(7) \end{gathered}$$
Step 6: Now, we have to subtract the expression (7) from the expression (6). Hence,
$$\begin{gathered} \Rightarrow 15x+25y-15x+9y=-30-72 \\ \Rightarrow 34y=-102 \\ \Rightarrow y=-{\displaystyle \frac{102}{34}} \\ \Rightarrow y=-3 \end{gathered}$$
Step 7: Now, to obtain the value of x we have to substitute the value of y in equation (4). Hence,
$$\begin{gathered} \Rightarrow 3x+5(-3)=-6 \\ \Rightarrow 3x=-6+15 \\ \Rightarrow x={\displaystyle \frac{9}{3}} \\ \Rightarrow x=3 \end{gathered}$$

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 $\bar{x}$
If the imaginary term i is multiplied with i or on squaring i means $(i{)}^2$ we will obtain the real term as -1.