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: Rationalize the denominators.
$\frac{2x}{1+3i} = \frac{2x(1-3i)}{(1+3i)(1-3i)} = \frac{2x - 6xi}{1 + 9} = \frac{2x - 6xi}{10}$.
$\frac{y}{1-2i} = \frac{y(1+2i)}{(1-2i)(1+2i)} = \frac{y + 2yi}{1 + 4} = \frac{y + 2yi}{5}$.


Step 2: Substitute into the equation.
$50 \left( \frac{2x - 6xi}{10} + \frac{y + 2yi}{5} \right) = 31 + 17i$
$5 (2x - 6xi) + 10 (y + 2yi) = 31 + 17i$
$(10x + 10y) + i(-30x + 20y) = 31 + 17i$


Step 3: Compare real and imaginary parts.
Real part: $10x + 10y = 31$ ---(1)
Imaginary part: $-30x + 20y = 17$ ---(2)


Step 4: Solve the system of equations.
Multiply (1) by 3: $30x + 30y = 93$.
Add to (2): $(30x + 30y) + (-30x + 20y) = 93 + 17$
$50y = 110 \implies y = \frac{11}{5} = 2.2$.
Substitute $y$ in (1): $10x + 10(2.2) = 31 \implies 10x + 22 = 31 \implies 10x = 9 \implies x = 0.9$.


Step 5: Calculate $10(x + 3y)$.
$10x + 30y = 9 + 30(\frac{11}{5}) = 9 + 6 \cdot 11 = 9 + 66 = 75$.
The answer is 75.