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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!

Given the equation, (2x - j5y) + 3(y + jx) = 8 - 7j. Find x.

Learn how to solve a complex equation with both real and imaginary parts by grouping terms and solving a system of linear equations. Suitable for students in Grades 10-12.

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