Question

Find the values of x and y which satisfy the following equation \[\dfrac{{\left( {1 + i} \right)x - 2i}}{{\left( {3 + i} \right)}} + \dfrac{{\left( {2 - 3i} \right)y + i}}{{\left( {3 - i} \right)}} = i\]
A. x = -1, y = 3
B. x = 3, y = -1
C. x = 0, y = 1
D. x = 1, y = 0

Answer 1

Hint: To find the value of x and y we need to solve the given equation and then compare the real and imaginary equation. We also need to know formulas for quadratic equation like, $( a^2 - b^2) = ( a + b) (a - b)$ also cross multiplication will be used in the above equation.

Complete step by step Solution:
Given \[\dfrac{{\left( {1 + i} \right)x - 2i}}{{\left( {3 + i} \right)}} + \dfrac{{\left( {2 - 3i} \right)y + i}}{{\left( {3 - i} \right)}} = i\]
\[\dfrac{{x\left( {x - 2} \right)i}}{{\left( {3 + i} \right)}} + \dfrac{{2y + \left( {1 - 3y} \right)i}}{{\left( {3 - i} \right)}} = i\]
In above equation (1) we will use Cross multiplication and $( a^2 - b^2) = ( a + b) (a - b)$
 \[\dfrac{{\left( {x\left( {x - 2} \right)i} \right)\left( {3 - i} \right) + \left( {2y + \left( {1 - 3y} \right)i} \right)\left( {3 + i} \right)}}{{9 - {i^2}}} = i\]
\[x\left( {3 - i} \right) + i(x - 2)\left( {3 - i} \right) + 2y\left( {3 + i} \right) + i\left( {1 - 3y} \right)\left( {3 + i} \right) = \left( {9 + 1} \right)i\]
\[3x - ix + i\left( {3x - ix - 6 + 2i} \right) + 6y + 2iy + i(3 + i - 9y - 3yi) = 10i\]
\[3x - ix + 3xi - {i^2}x - 6i + 2{i^2} + 6y + 2iy + 3i + {i^2} - 9yi - 3y{i^2} = 10i\]
\[3x - ix + 3xi + x - 6i - 2 + 6y + 2iy + 3i - 1 - 9yi - 3y = 10i\]
$4x + 9y - 3 + 2xi - 7yi - 13i = 0$
$4x + 9y - 3 + \left( {2x - 7y - 13} \right)i = 0$
Comparing real part and imaginary part, we get
\[4x + 9y - 3 = 0\]
\[2x - 7y - 13 = 0\]
Solving the above equations (2) and (3), we get

\[ \Rightarrow 23y = - 23\]
\[ \Rightarrow y = - 1\]

Substitute y in (1)
\[4x + 9\left( { - 1} \right) - 3 = 0\]
$ \Rightarrow x = 3$

Hence, the correct option is B.

Note: To solve this type of question taking the given option into consideration seems easy to find answers but after careful consideration of the equation imaginary and real equations will be finally considered. Also knowing basic quadratic formulas can help in simplifying the equation.