Question

The value of x and y in (3y-2)+i(7-2x)=0

• A. x=\(\frac{2}{7}\), y=\(\frac{2}{3}\)
• B. x=\(\frac{7}{2}\), y=\(\frac{3}{2}\)
• C. x=\(\frac{2}{7}\), y=\(\frac{3}{2}\)
• D. x=\(\frac{7}{2}\), y=\(\frac{2}{3}\)

Answer 1

The Correct Option is D

To find the values of \(x\) and \(y\) that satisfy the complex equation \((3y - 2) + i(7 - 2x) = 0\), we'll analyze both the real and imaginary parts separately.

1. Given Equation:
\[ (3y - 2) + i(7 - 2x) = 0 + 0i \]

2. Equating Real and Imaginary Parts:
For a complex number to be zero, both its real and imaginary parts must be zero: \[ \begin{cases} \text{Real part: } 3y - 2 = 0 \\ \text{Imaginary part: } 7 - 2x = 0 \end{cases} \]

3. Solving for \(y\):
\[ 3y - 2 = 0 \\ 3y = 2 \\ y = \frac{2}{3} \]

4. Solving for \(x\):
\[ 7 - 2x = 0 \\ 2x = 7 \\ x = \frac{7}{2} \]

5. Verification:
Substituting \(x = \frac{7}{2}\) and \(y = \frac{2}{3}\) back into the original equation: \[ (3 \cdot \tfrac{2}{3} - 2) + i(7 - 2 \cdot \tfrac{7}{2}) = (2 - 2) + i(7 - 7) = 0 + 0i \] Which satisfies the equation.

6. Comparing with Given Options:
The correct solution matches with the option: \[ x = \frac{7}{2}, \quad y = \frac{2}{3} \]

Final Answer:
The correct values are \(\boxed{x = \dfrac{7}{2}}\) and \(\boxed{y = \dfrac{2}{3}}\).