https://file.aiquickdraw.com/mathbot/file/1724177080113c5gz36b7.jpg

To solve the equation provided in the image, we can proceed step by step. The equation is:

\[
\frac{x - 2i}{2 + i} + \frac{2 - yi}{1 - i} = 3 - 2i
\]

### Step 1: Simplify the fractions

First, simplify the two fractions by multiplying the numerator and the denominator by the conjugate of the denominator.

#### For the first fraction:
\[
\frac{x - 2i}{2 + i} \times \frac{2 - i}{2 - i} = \frac{(x - 2i)(2 - i)}{(2 + i)(2 - i)}
\]
The denominator simplifies as:
\[
(2 + i)(2 - i) = 4 - i^2 = 4 + 1 = 5
\]
The numerator expands as:
\[
(x - 2i)(2 - i) = 2x - xi - 4i - 2i^2 = 2x - xi - 4i + 2 = (2x + 2) - (x + 4)i
\]
So, the first fraction simplifies to:
\[
\frac{(2x + 2) - (x + 4)i}{5} = \frac{2x + 2}{5} - \frac{x + 4}{5}i
\]

#### For the second fraction:
\[
\frac{2 - yi}{1 - i} \times \frac{1 + i}{1 + i} = \frac{(2 - yi)(1 + i)}{(1 - i)(1 + i)}
\]
The denominator simplifies as:
\[
(1 - i)(1 + i) = 1 - i^2 = 1 + 1 = 2
\]
The numerator expands as:
\[
(2 - yi)(1 + i) = 2 + 2i - yi - yi^2 = 2 + 2i - yi + y = (2 + y) + (2 - y)i
\]
So, the second fraction simplifies to:
\[
\frac{(2 + y) + (2 - y)i}{2} = \frac{2 + y}{2} + \frac{2 - y}{2}i
\]

### Step 2: Combine the simplified expressions
The equation now becomes:
\[
\frac{2x + 2}{5} - \frac{x + 4}{5}i + \frac{2 + y}{2} + \frac{2 - y}{2}i = 3 - 2i
\]
Group the real and imaginary parts:
\[
\left(\frac{2x + 2}{5} + \frac{2 + y}{2}\right) + \left(-\frac{x + 4}{5} + \frac{2 - y}{2}\right)i = 3 - 2i
\]

### Step 3: Set up the system of equations
Since the real parts and imaginary parts on both sides must be equal, we have:
1. **Real part equation:**
   \[
   \frac{2x + 2}{5} + \frac{2 + y}{2} = 3
   \]
2. **Imaginary part equation:**
   \[
   -\frac{x + 4}{5} + \frac{2 - y}{2} = -2
   \]

### Step 4: Solve the system of equations
1. **Real part equation:**
   \[
   \frac{2x + 2}{5} + \frac{2 + y}{2} = 3
   \]
   Multiply the entire equation by 10 to eliminate the fractions:
   \[
   2(2x + 2) + 5(2 + y) = 30
   \]
   Simplify:
   \[
   4x + 4 + 10 + 5y = 30
   \]
   \[
   4x + 5y + 14 = 30 \implies 4x + 5y = 16
   \]
   \[
   \text{Equation 1: } 4x + 5y = 16
   \]

2. **Imaginary part equation:**
   \[
   -\frac{x + 4}{5} + \frac{2 - y}{2} = -2
   \]
   Multiply the entire equation by 10 to eliminate the fractions:
   \[
   -2(x + 4) + 5(2 - y) = -20
   \]
   Simplify:
   \[
   -2x - 8 + 10 - 5y = -20
   \]
   \[
   -2x - 5y + 2 = -20 \implies -2x - 5y = -22
   \]
   \[
   \text{Equation 2: } 2x + 5y = 22
   \]

### Step 5: Solve the system of linear equations
Now solve the equations:
\[
4x + 5y = 16 \quad \text{(Equation 1)}
\]
\[
2x + 5y = 22 \quad \text{(Equation 2)}
\]

Subtract Equation 2 from Equation 1:
\[
(4x + 5y) - (2x + 5y) = 16 - 22
\]
\[
2x = -6 \implies x = -3
\]

Substitute \(x = -3\) into Equation 1:
\[
4(-3) + 5y = 16
\]
\[
-12 + 5y = 16 \implies 5y = 28 \implies y = \frac{28}{5} = 5.6
\]

### Solution:
\[
x = -3, \quad y = 5.6
\]

Would you like more details on this process or have any questions?

Here are 5 related questions for practice:
1. Solve the equation \(\frac{3 + 2i}{1 + i} + \frac{x - 5i}{2 - i} = 4 + 3i\).
2. Simplify \(\frac{7 - 3i}{1 - 2i}\).
3. Find the magnitude and argument of the complex number \(3 + 4i\).
4. How do you multiply two complex numbers \(a + bi\) and \(c + di\)?
5. Solve for \(z\) if \(z = \frac{1}{1 + i}\).

**Tip:** Always simplify complex fractions by multiplying the numerator and the denominator by the conjugate of the denominator.

Learn how to solve a complex equation involving fractions and variables in complex numbers. This detailed step-by-step solution covers complex number operations, fraction simplification, and solving systems of equations. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation