We are given the equation:

$(2x - j5y) + 3(y + jx) = 8 - 7j$

Our goal is to find $x$, where $j$ represents the imaginary unit (i.e., $j = \sqrt{-1}$).

Step 1: Expand the terms on the left side

Let's expand both groups:

$2x - j5y + 3(y + jx)$

Distribute the 3 across $(y + jx)$:

$2x - j5y + 3y + 3jx$

This simplifies to:

$(2x + 3jx) + (3y - j5y)$

Step 2: Group real and imaginary parts

Group the real parts (involving $x$ and $y$) and the imaginary parts (involving $j$) separately:

Real part:

$2x + 3y$

Imaginary part:

$j(3x - 5y)$

So, the equation becomes:

$(2x + 3y) + j(3x - 5y) = 8 - 7j$

Step 3: Compare real and imaginary parts

We now equate the real and imaginary parts on both sides:

Real part: $2x + 3y = 8$

Imaginary part: $3x - 5y = -7$

Step 4: Solve the system of equations

We have the system:

1. $2x + 3y = 8$
2. $3x - 5y = -7$

We can solve this system using substitution or elimination. I'll use elimination here.

Multiply the first equation by 3 and the second by 2:

1. $6x + 9y = 24$
2. $6x - 10y = -14$

Now subtract equation 2 from equation 1:

$(6x + 9y) - (6x - 10y) = 24 - (-14)$

This simplifies to:

$19y = 38$

Solving for $y$:

$y = 2$

Step 5: Substitute $y = 2$ into one of the original equations

Substitute $y = 2$ into the first equation $2x + 3y = 8$:

$2x + 3(2) = 8$

This simplifies to:

$2x + 6 = 8$

Solving for $x$:

$2x = 2 \quad \Rightarrow \quad x = 1$

Final Answer:

The value of $x$ is $1$.

Would you like more details on any step, or do you have any further questions?

Here are five related questions:

1. How would you solve the system of equations using substitution?
2. What happens if $y$ is negative in this type of equation?
3. How does the presence of the imaginary unit $j$ affect the equation-solving process?
4. Can you graph the system of equations in the real-imaginary plane?
5. How would this problem change if the constants on the right side were complex numbers with different magnitudes?

Tip: Always group real and imaginary components separately when solving equations involving complex numbers!