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
Solution
1. Break Down the Problem
We need to solve the equation: where . This requires simplifying both sides and isolating the variables and .
2. Relevant Concepts
- Substitute .
- Use properties of complex numbers to simplify the equation.
- Set the real and imaginary parts equal to isolate the variables.
3. Analysis and Detail
Step 1: Substitute
Substituting into the equation:
This simplifies to:
Step 2: Simplifying the numerator
So, the numerator becomes:
This gives:
Step 3: Simplifying the denominator
For the denominator: So, it becomes:
Step 4: Set up the equation
Now, rewrite the equation in simpler form:
Step 5: Cross-multiply
Cross-multiplying gives:
Step 6: Equate real and imaginary parts
We can now separate real and imaginary components:
- Real part:
- Imaginary part:
4. Verify and Summarize
Solving the imaginary part:
Now substitute back into the real part: This simplifies to: Multiply through by 3: Combine terms:
Final Answer
Thus, the values of and that satisfy the equation are:
Similar Questions
For z=x+iy𝑧=𝑥+𝑖𝑦 find the values x𝑥 and y𝑦 satisfying 2z1+i−2zi=52+i2𝑧1+𝑖−2𝑧𝑖=52+𝑖.
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