Question

Let \(3x + 2iy - ix + 5y = 7 + 5i\), then values of \(x\) and \(y\) respectively are given by

• A. -1, 2
• B. 2, -1
• C. -8, 3
• D. 1, -2

Hint:

To quickly verify your answer, substitute the options back into the imaginary part equation (\( 2y - x = 5 \)):
- Option (A): \( 2(2) - (-1) = 4 + 1 = 5 \). This is correct!
- Option (B): \( 2(-1) - 2 = -4 \neq 5 \).
This simple substitution saves time and prevents algebraic mistakes.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem requires solving for real variables \( x \) and \( y \) by equating two complex expressions.
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

Step 2: Detailed Explanation:
Let us separate the real and imaginary components on both sides of the given equation:
\[ 3x + 2iy - ix + 5y = 7 + 5i \] - Group the terms on the left side:
- Real terms (terms without the imaginary unit \( i \)):
\[ 3x + 5y \] - Imaginary terms (terms containing \( i \)):
\[ 2iy - ix = (2y - x)i \] So, the left side can be written as:
\[ (3x + 5y) + (2y - x)i \] - Equate the real and imaginary parts to the right side (\( 7 + 5i \)):
- Equating real parts:
\[ 3x + 5y = 7 \quad \text{--- (Equation 1)} \] - Equating imaginary parts:
\[ 2y - x = 5 \implies x = 2y - 5 \quad \text{--- (Equation 2)} \] - Solve the system of linear equations:
Substitute the expression for \( x \) from Equation 2 into Equation 1:
\[ 3(2y - 5) + 5y = 7 \] \[ 6y - 15 + 5y = 7 \] \[ 11y - 15 = 7 \] \[ 11y = 22 \implies y = 2 \] Now, substitute \( y = 2 \) back into Equation 2 to find \( x \):
\[ x = 2(2) - 5 = 4 - 5 = -1 \] Thus, we find \( x = -1 \) and \( y = 2 \).

Step 3: Final Answer:
The values of \( x \) and \( y \) are \( -1 \) and \( 2 \), respectively.
Therefore, the correct choice is Option (A).