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

Steps to Solve

1. Determine the conjugate of the complex number

The conjugate of the complex number $-6 - 24i$ is found by changing the sign of the imaginary part. Therefore,

$$ \text{Conjugate} = -6 + 24i $$

2. Set up the equation

We need to find $x$ and $y$ such that the product of $(x - iy)$ and $(3 + 5i)$ equals the conjugate we found:

$$ (x - iy)(3 + 5i) = -6 + 24i $$

3. Expand the left side

Use the distributive property (FOIL method) to multiply:

$$ (x - iy)(3 + 5i) = 3x + 5xi - 3iy - 5y $$

Combining like terms, we have:

$$ = (3x + 5y) + (5x - 3y)i $$

4. Equate real and imaginary parts

Now, we set the real part equal to the real part of the conjugate, and the imaginary part equal to the imaginary part:

$$ 3x + 5y = -6 $$

$$ 5x - 3y = 24 $$

5. Solve the system of equations

We have a system of linear equations:

1. $3x + 5y = -6$

2. $5x - 3y = 24$

We can solve for $x$ and $y$ using substitution or elimination. For example, we’ll use substitution for $y$ from equation (1):

From (1):

$$ 5y = -6 - 3x \implies y = \frac{-6 - 3x}{5} $$

Now substitute $y$ into equation (2):

$$ 5x - 3(\frac{-6 - 3x}{5}) = 24 $$

Multiply through by 5 to clear the fraction:

$$ 25x + 18 + 9x = 120 $$

Combine like terms:

$$ 34x + 18 = 120 $$

Solving for $x$ gives:

$$ 34x = 102 \implies x = 3 $$

Now substitute $x = 3$ back to find $y$:

$$ 3(3) + 5y = -6 \implies 9 + 5y = -6 \implies 5y = -15 \implies y = -3 $$