StudyGPT Just Leveled Up โ€“ Say Hello to Knowee! ๐Ÿš€
Knowee
Questions
Features
Study Tools

For z=x+iy๐‘ง=๐‘ฅ+๐‘–๐‘ฆ find the values x๐‘ฅ and y๐‘ฆ satisfying 2z1+iโˆ’2zi=52+i2๐‘ง1+๐‘–โˆ’2๐‘ง๐‘–=52+๐‘–.

Question

๐Ÿง Not the exact question you are looking for?Go ask a question

Solution

1. Break Down the Problem

We need to solve the equation: 2z1+iโˆ’2zi52+i2z=1 2z \frac{1+i-2zi}{52+i2z} = 1 where z=x+iy z = x + iy . This requires simplifying both sides and isolating the variables x x and y y .

2. Relevant Concepts

  1. Substitute z=x+iy z = x + iy .
  2. Use properties of complex numbers to simplify the equation.
  3. Set the real and imaginary parts equal to isolate the variables.

3. Analysis and Detail

Step 1: Substitute z z

Substituting z=x+iy z = x + iy into the equation:

2(x+iy)1+iโˆ’2i(x+iy)52+i2(x+iy)=1 2(x + iy) \frac{1+i-2i(x+iy)}{52+i2(x+iy)} = 1

This simplifies to:

Step 2: Simplifying the numerator

1+iโˆ’2i(x+iy)=1+iโˆ’2ix+2y=(1+2y)+i(1โˆ’2x) 1 + i - 2i(x + iy) = 1 + i - 2ix + 2y = (1 + 2y) + i(1 - 2x)

So, the numerator becomes: 2(x+iy)((1+2y)+i(1โˆ’2x)) 2(x + iy) \left( (1 + 2y) + i(1 - 2x) \right)

This gives: 2x(1+2y)+2iy(1+2y)+2i(1โˆ’2x)+2y(1โˆ’2x) 2x(1 + 2y) + 2iy(1 + 2y) + 2i(1 - 2x) + 2y(1 - 2x)

Step 3: Simplifying the denominator

For the denominator: 52+i2(x+iy)=52+2ixโˆ’2y 52 + i2(x + iy) = 52 + 2ix - 2y So, it becomes: 52+2ixโˆ’2y 52 + 2ix - 2y

Step 4: Set up the equation

Now, rewrite the equation in simpler form: 2x(1+2y)+2i(1โˆ’2x)52โˆ’2y+2ix=1 \frac{2x(1 + 2y) + 2i(1 - 2x)}{52 - 2y + 2ix} = 1

Step 5: Cross-multiply

Cross-multiplying gives: 2x(1+2y)+2i(1โˆ’2x)=52โˆ’2y+2ix 2x(1 + 2y) + 2i(1 - 2x) = 52 - 2y + 2ix

Step 6: Equate real and imaginary parts

We can now separate real and imaginary components:

  1. Real part: 2x(1+2y)=52โˆ’2y 2x(1 + 2y) = 52 - 2y
  2. Imaginary part: 2(1โˆ’2x)=2x 2(1 - 2x) = 2x

4. Verify and Summarize

Solving the imaginary part: 2โˆ’4x=2xโ€…โ€ŠโŸนโ€…โ€Š2=6xโ€…โ€ŠโŸนโ€…โ€Šx=13 2 - 4x = 2x \implies 2 = 6x \implies x = \frac{1}{3}

Now substitute x=13 x = \frac{1}{3} back into the real part: 2โ‹…13(1+2y)=52โˆ’2y 2 \cdot \frac{1}{3}(1 + 2y) = 52 - 2y This simplifies to: 23(1+2y)=52โˆ’2y \frac{2}{3}(1 + 2y) = 52 - 2y Multiply through by 3: 2(1+2y)=156โˆ’6yโ€…โ€ŠโŸนโ€…โ€Š2+4y=156โˆ’6y 2(1 + 2y) = 156 - 6y \implies 2 + 4y = 156 - 6y Combine terms: 10y=154โ€…โ€ŠโŸนโ€…โ€Šy=15410=15.4 10y = 154 \implies y = \frac{154}{10} = 15.4

Final Answer

Thus, the values of x x and y y that satisfy the equation are: x=13,y=15.4 x = \frac{1}{3}, \quad y = 15.4

This problem has been solved

Upgrade your grade with Knowee

Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.