The problem in the image asks to solve for the real numbers $x$ and $y$ given the equation:

$(x + yi) + (1 + 5i) = (x + yi)(1 + 5i)$

Step 1: Simplify the Left-Hand Side (LHS)

The left-hand side is simply the addition of two complex numbers: $(x + yi) + (1 + 5i) = (x + 1) + (y + 5)i$

Step 2: Expand the Right-Hand Side (RHS)

The right-hand side involves multiplying two complex numbers: $(x + yi)(1 + 5i)$

Using the distributive property (FOIL method):

$= x(1) + x(5i) + yi(1) + yi(5i)$

$= x + 5xi + yi + 5y(i^2)$

Since $i^2 = -1$, this becomes:

$= x + 5xi + yi - 5y$

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

Step 3: Equate the Real and Imaginary Parts

Now we have the equation:

$(x + 1) + (y + 5)i = (x - 5y) + (5x + y)i$

Equating the real and imaginary parts separately:

• Real part: $x + 1 = x - 5y$ Simplifying: $1 = -5y \quad \Rightarrow \quad y = -\frac{1}{5}$

• Imaginary part: $y + 5 = 5x + y$ Simplifying: $5 = 5x \quad \Rightarrow \quad x = 1$

Final Solution:

The real numbers $x$ and $y$ are: $x = 1, \quad y = -\frac{1}{5}$

Would you like further clarification or more details?

Here are some related questions:

1. How do you solve equations involving complex numbers in general?
2. Can you explain the properties of imaginary numbers like $i^2 = -1$?
3. How do you perform the multiplication of two complex numbers?
4. What is the geometric interpretation of complex number multiplication?
5. How does equating real and imaginary parts help in solving complex equations?

Tip: When solving complex equations, remember to treat real and imaginary parts separately by comparing coefficients!