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For z=x+iy𝑧=𝑥+𝑖𝑦 find the values x𝑥 and y𝑦 satisfying 2z1+i−2zi=52+i2𝑧1+𝑖−2𝑧𝑖=52+𝑖.

Question

For z=x+iy find the values x and y satisfying

2z1+i2zi=52+i 2z \frac{1+i}{-2zi} = \frac{5}{2+i}
2z1+i2z=52+i 2z \frac{1+i}{-2z} = \frac{5}{2+i}

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Solution

1. Break Down the Problem

We need to solve the equation: 2z1+i2zi52+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+i2i(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+i2i(x+iy)=1+i2ix+2y=(1+2y)+i(12x) 1 + i - 2i(x + iy) = 1 + i - 2ix + 2y = (1 + 2y) + i(1 - 2x)

So, the numerator becomes: 2(x+iy)((1+2y)+i(12x)) 2(x + iy) \left( (1 + 2y) + i(1 - 2x) \right)

This gives: 2x(1+2y)+2iy(1+2y)+2i(12x)+2y(12x) 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+2ix2y 52 + i2(x + iy) = 52 + 2ix - 2y So, it becomes: 52+2ix2y 52 + 2ix - 2y

Step 4: Set up the equation

Now, rewrite the equation in simpler form: 2x(1+2y)+2i(12x)522y+2ix=1 \frac{2x(1 + 2y) + 2i(1 - 2x)}{52 - 2y + 2ix} = 1

Step 5: Cross-multiply

Cross-multiplying gives: 2x(1+2y)+2i(12x)=522y+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)=522y 2x(1 + 2y) = 52 - 2y
  2. Imaginary part: 2(12x)=2x 2(1 - 2x) = 2x

4. Verify and Summarize

Solving the imaginary part: 24x=2x    2=6x    x=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: 213(1+2y)=522y 2 \cdot \frac{1}{3}(1 + 2y) = 52 - 2y This simplifies to: 23(1+2y)=522y \frac{2}{3}(1 + 2y) = 52 - 2y Multiply through by 3: 2(1+2y)=1566y    2+4y=1566y 2(1 + 2y) = 156 - 6y \implies 2 + 4y = 156 - 6y Combine terms: 10y=154    y=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

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