Complex numbers-I don't really understand it

We are asked to find the complex number $z$ that satisfies the equation:

 

$$(1+i)z - 2z^* = -11 + 25i$$

 

where $z^*$ denotes the conjugate of $z$. Let $z = x + yi$, where $x$ and $y$ are real numbers. Then, the conjugate $z^*$ is given by:

 

$$z^* = x - yi$$

 

### Step 1: Substitute $z = x + yi$ and $z^* = x - yi$ into the equation

 

Substituting these into the equation, we get:

 

$$(1+i)(x + yi) - 2(x - yi) = -11 + 25i$$

 

### Step 2: Expand the terms

 

First, expand $(1+i)(x + yi)$:

 

$$(1+i)(x + yi) = 1 \cdot x + 1 \cdot yi + i \cdot x + i \cdot yi = x + yi + xi + i^2y = (x - y) + i(x + y)$$

 

Next, expand $-2(x - yi)$:

 

$$-2(x - yi) = -2x + 2yi$$

 

Substituting these into the equation:

 

$$(x - y + 2y)i + (x - 2x - y) = -11 + 25i$$

 

### Step 3: Combine like terms

 

Combining the real parts and imaginary parts, we have:

 

$$(x - y - 2x) + (2y + x + 2yi) = -11 + 25i$$

 

This simplifies to:

 

$$-x - y + 2yi + (x - 2x) + (y + y)i = -11 + 25i$$

 

### Step 4: Equate real and imaginary parts

 

Now, equate the real and imaginary parts of the equation:

 

For the real part:

 

$$x - 2y = -11$$

 

For the imaginary part:

 

$$x + 2y = 25$$

 

### Step 5: Solve the system of equations

 

We now solve the system of equations:

 

$$x - 2y = -11$$

$$x + 2y = 25$$

 

Add these two equations:

 

$$(x - 2y) + (x + 2y) = -11 + 25$$

 

This simplifies to:

 

$$2x = 14$$

 

So:

 

$$x = 7$$

 

Substitute $x = 7$ into one of the original equations:

 

$$7 - 2y = -11$$

 

Solve for $y$:

$$-2y = -18 \quad \Rightarrow \quad y = 9$$

 

### Step 6: Write the complex number $z$

 

The complex number $z$ is:

 

$$z = x + yi = 7 + 9i$$

 

Thus, the solution is:

 

$$\boxed{7 + 9i}$$