If `8x+i(2x-y)=3 -8i` and `x,y in R` then the values of x and y are

Options

A. `x= (3)/(8) , y = (35)/(4)`
B. `x=- (3)/(8), y =(35)/(4)`
C. `x= (3)/(8), y =-(35)/(4)`
D. ` x= -(3)/(8), y=-(35)/(4)`

Answer

To solve the equation \( 8x + i(2x - y) = 3 - 8i \), we can separate the real and imaginary parts.

### Step 1: Separate Real and Imaginary Parts
We can rewrite the equation as:
\[ 8x + i(2x - y) = 3 - 8i \]
From this, we can equate the real parts and the imaginary parts:
- Real part: \( 8x = 3 \)
- Imaginary part: \( 2x - y = -8 \)

### Step 2: Solve for \( x \)
From the real part equation \( 8x = 3 \), we can solve for \( x \):
\[ x = \frac{3}{8} \]

### Step 3: Substitute \( x \) into the Imaginary Part Equation
Now, we substitute \( x = \frac{3}{8} \) into the imaginary part equation \( 2x - y = -8 \):
\[ 2\left(\frac{3}{8}\right) - y = -8 \]
This simplifies to:
\[ \frac{6}{8} - y = -8 \]
\[ \frac{3}{4} - y = -8 \]

### Step 4: Solve for \( y \)
Now, we can isolate \( y \):
\[ -y = -8 - \frac{3}{4} \]
To combine the terms on the right, we convert \(-8\) into a fraction:
\[ -8 = -\frac{32}{4} \]
So,
\[ -y = -\frac{32}{4} - \frac{3}{4} = -\frac{35}{4} \]
Multiplying both sides by -1 gives:
\[ y = \frac{35}{4} \]

### Final Result
Thus, the values of \( x \) and \( y \) are:
\[ x = \frac{3}{8}, \quad y = \frac{35}{4} \]

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