Find the real numbers x and y if (x - iy)(3 + 5i) is the conjugate of -6 - 24i.
Understand the Problem
The question is asking us to find the real numbers x and y such that the product of the complex number (x - iy) and (3 + 5i) is equal to the conjugate of the complex number -6 - 24i. To solve this, we will first determine the conjugate of -6 - 24i, which is -6 + 24i, and then solve for x and y by equating the product of the complex numbers.
Answer
$x = 3, y = -3$
Answer for screen readers
The values of $x$ and $y$ are:
$x = 3, y = -3$
Steps to Solve
- 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,
$$ \text{Conjugate} = -6 + 24i $$
- 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:
$$ (x - iy)(3 + 5i) = -6 + 24i $$
- 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:
$$ = (3x + 5y) + (5x - 3y)i $$
- 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:
$$ 3x + 5y = -6 $$
$$ 5x - 3y = 24 $$
- Solve the system of equations
We have a system of linear equations:
-
$3x + 5y = -6$
-
$5x - 3y = 24$
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 $y$:
$$ 3(3) + 5y = -6 \implies 9 + 5y = -6 \implies 5y = -15 \implies y = -3 $$
The values of $x$ and $y$ are:
$x = 3, y = -3$
More Information
When multiplying complex numbers, remember to separate the real and imaginary parts for easier calculations. The operations will often yield a set of linear equations.
Tips
- Forgetting to take the conjugate correctly.
- Not distributing correctly while multiplying complex numbers.
- Forgetting to separate and equate real and imaginary parts.
