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We are tasked with finding real values for \( x \) and \( y \) such that:

\[
(x - iy)(3 + 5i)
\]
is the conjugate of \( -6 - 24i \).

### Step 1: Expand the product \( (x - iy)(3 + 5i) \)

Using the distributive property:
\[
(x - iy)(3 + 5i) = x(3 + 5i) - iy(3 + 5i)
\]
Expand both terms:
\[
x(3 + 5i) = 3x + 5ix
\]
\[
-iy(3 + 5i) = -3iy - 5i^2y = -3iy + 5y \quad (\text{since } i^2 = -1)
\]

Now, combine the real and imaginary parts:
\[
(3x + 5y) + i(5x - 3y)
\]

### Step 2: Equate the result with the conjugate of \( -6 - 24i \)

The conjugate of \( -6 - 24i \) is:
\[
-6 + 24i
\]

Thus, we have the equation:
\[
(3x + 5y) + i(5x - 3y) = -6 + 24i
\]

### Step 3: Compare real and imaginary parts

For the real parts:
\[
3x + 5y = -6
\]

For the imaginary parts:
\[
5x - 3y = 24
\]

### Step 4: Solve the system of equations

1. \( 3x + 5y = -6 \)
2. \( 5x - 3y = 24 \)

#### Solving by substitution or elimination:

Multiply the first equation by 3 and the second equation by 5 to eliminate \( y \):
\[
9x + 15y = -18 \quad (3 \times \text{Eq. 1})
\]
\[
25x - 15y = 120 \quad (5 \times \text{Eq. 2})
\]

Add the two equations:
\[
(9x + 25x) + (15y - 15y) = -18 + 120
\]
\[
34x = 102
\]
\[
x = \frac{102}{34} = 3
\]

Substitute \( x = 3 \) into the first equation:
\[
3(3) + 5y = -6
\]
\[
9 + 5y = -6
\]
\[
5y = -6 - 9 = -15
\]
\[
y = \frac{-15}{5} = -3
\]

### Final Answer:
The values of \( x \) and \( y \) are:
\[
x = 3, \quad y = -3
\]

Would you like more details on any part of this process? Here are five related questions:

1. What is the geometric meaning of complex conjugates in the complex plane?
2. How does expanding and simplifying complex numbers relate to binomials in algebra?
3. What are the general properties of multiplying complex numbers?
4. Could the method of solving systems of equations be applied to larger systems with more variables?
5. How do real and imaginary parts affect the behavior of complex numbers in equations?

**Tip**: When working with complex numbers, remember that the imaginary unit \( i \) satisfies \( i^2 = -1 \), which is crucial for simplifying products of complex numbers.

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

Step-by-step solution for finding real values of x and y such that (x - iy)(3 + 5i) equals the conjugate of -6 - 24i. Learn to expand, compare real and imaginary parts, and solve a system of equations. Suitable for students in Grades 11-12.

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