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

Answer

To find the real numbers \( x \) and \( y \) such that \( (x - yi)(3 + 5i) \) is the conjugate of \( -6 - 24i \), we can follow these steps:

### Step 1: Write the conjugate of \(-6 - 24i\)
The conjugate of a complex number \( a + bi \) is \( a - bi \). Therefore, the conjugate of \( -6 - 24i \) is:
\[ -6 + 24i \]

### Step 2: Expand the expression \( (x - yi)(3 + 5i) \)
Now we will expand the left-hand side:
\[ (x - yi)(3 + 5i) = x \cdot 3 + x \cdot 5i - yi \cdot 3 - yi \cdot 5i \]
This simplifies to:
\[ 3x + 5xi - 3yi - 5y(-1) \]
Since \( i^2 = -1 \), we can rewrite it as:
\[ 3x + 5xi + 5y - 3yi \]
Combining the real and imaginary parts gives us:
\[ (3x + 5y) + (5x - 3y)i \]

### Step 3: Set the expanded expression equal to the conjugate
Now we set the expanded expression equal to the conjugate we found in Step 1:
\[ (3x + 5y) + (5x - 3y)i = -6 + 24i \]

### Step 4: Create a system of equations
From the equality of the real and imaginary parts, we can create two equations:
1. \( 3x + 5y = -6 \) (Real part)
2. \( 5x - 3y = 24 \) (Imaginary part)

### Step 5: Solve the system of equations
We will solve these equations simultaneously.

**From Equation 1:**
\[ 3x + 5y = -6 \implies 3x = -6 - 5y \implies x = \frac{-6 - 5y}{3} \]

**Substituting \( x \) into Equation 2:**
\[ 5\left(\frac{-6 - 5y}{3}\right) - 3y = 24 \]
Multiplying through by 3 to eliminate the fraction:
\[ 5(-6 - 5y) - 9y = 72 \]
This simplifies to:
\[ -30 - 25y - 9y = 72 \]
Combining like terms:
\[ -30 - 34y = 72 \]
Adding 30 to both sides:
\[ -34y = 102 \]
Dividing by -34:
\[ y = -3 \]

### Step 6: Substitute \( y \) back to find \( x \)
Now we substitute \( y = -3 \) back into Equation 1:
\[ 3x + 5(-3) = -6 \]
This simplifies to:
\[ 3x - 15 = -6 \]
Adding 15 to both sides:
\[ 3x = 9 \]
Dividing by 3:
\[ x = 3 \]

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