To determine the real values of $x$ and $y$, we equate the real and imaginary parts of the given complex numbers:

Given equation:

$x + 4 + i = 10 - (2y - 7)i$

Step 1: Split into real and imaginary parts

• Real part: $x + 4$ must equal $10$.
• Imaginary part: $1$ must equal $-(2y - 7)$.

Step 2: Solve for $x$ and $y$

Real part:

$x + 4 = 10$ $x = 10 - 4 = 6$

Imaginary part:

$1 = -(2y - 7)$ $1 = -2y + 7$ $-2y = 1 - 7$ $-2y = -6$ $y = \frac{-6}{-2} = 3$

Final solution:

$x = 6, \quad y = 3$

Would you like to see further details or clarifications? Here are 5 related questions you might explore:

1. How can we verify the solution for $x$ and $y$ in this problem?
2. What are the general properties of equating complex numbers?
3. How do the real and imaginary components behave in more complicated equations?
4. Can this method be extended to systems of complex equations?
5. What happens if there are no imaginary components in a similar problem?

Tip: When solving equations involving complex numbers, always separate real and imaginary parts before proceeding to simplify.