Question

If $50 \left( \frac{2x}{1 + 3i} + \frac{y}{1 - 2i} \right) = 31 + 17i$ where $x, y \in R$ & $i = \sqrt{-1}$ then value of $10(x + 3y)$ is ________

Answer 1

Step 1: Understanding the Concept: 
We simplify the complex fraction by multiplying the numerator and denominator by the conjugate of the denominator. 
Step 2: Key Formula or Approach: 
\( \frac{1}{a+ib} = \frac{a-ib}{a^2+b^2} \). 
Step 3: Detailed Explanation: 
\( \frac{2x}{1+3i} = \frac{2x(1-3i)}{10} = \frac{x-3xi}{5} \). 
\( \frac{y}{1-2i} = \frac{y(1+2i)}{5} \). 
Equation becomes: 
\[ 50 \left( \frac{x - 3xi + y + 2yi}{5} \right) = 31 + 17i \] 
\[ 10(x+y) + i(20y-30x) = 31 + 17i \] 
Equating real parts: \( 10(x+y) = 31 \implies 10x + 10y = 31 \) …….. (1). 
Equating imaginary parts: \( 20y - 30x = 17 \) ……….. (2). 
From (1), \( 30x + 30y = 93 \). 
Adding this to (2): \( 50y = 110 \implies y = 2.2 \). 
From (1), \( 10x = 31 - 22 = 9 \). 
Required value: \( 10x + 30y = 9 + 3(22) = 9 + 66 = 75 \). 
Step 4: Final Answer: 
The value of \( 10(x+3y) \) is 75.