To solve the equation
\[ \frac{2}{i} - 2y + i^{49} = x + \frac{I}{I} + y, \]
we can first simplify it step by step.
Step 1: Simplify the components
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Calculate \( \frac{2}{i} \): \[ \frac{2}{i} = 2 \cdot \frac{-i}{-i} = -2i. \]
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Calculate \( i^{49} \): The powers of \( i \) cycle every 4:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)
To find \( i^{49} \): \[ 49 \mod 4 = 1 \quad \text{(since } 49 = 48 + 1 \text{)} \] Thus, \( i^{49} = i \).
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Evaluate \( \frac{I}{I} \): Assuming \( I \) refers to the imaginary unit \( i \): \[ \frac{i}{i} = 1. \] If \( I \) stands for something else, please clarify.
Step 2: Substitute into the equation
Now we substitute all simplified parts back into the equation: \[ -2i - 2y + i = x + 1 + y. \]
Combine like terms on the left-hand side: \[ (-2i + i) - 2y = x + 1 + y \quad \Rightarrow \quad -i - 2y = x + 1 + y. \]
Step 3: Isolate variables
Now, we can isolate \( x \) and \( y \): \[ -i - 2y = x + 1 + y. \]
Rearranging gives: \[ x = -i - 2y - 1 - y \quad \Rightarrow \quad x = -i - 3y - 1. \]
Conclusion
So the relationship between \( x \) and \( y \) is given by: \[ x = -1 - i - 3y. \]
This equation tells us how \( x \) and \( y \) are related. If you have specific values for \( y \), you can substitute them in to find corresponding values of \( x \).
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