Question

If x and y are complex numbers then system of equations $\left( {1 + i} \right)x + \left( {1 - i} \right)y = 1,\,\,2xi + 2y = 1 + i$ has
(A) unique solution
(B) no solution
(C) infinite number of solution
(D) none of these

Answer 1

Hint: Writing the given complex equation in standard form then solving equations obtained on comparing to get a number of solutions.
Formulas used: $x + iy = a + ib$ comparing real part with real part and imaginary part with imaginary part

Complete step by step solution.
Given, equations are \[\left( {1 + i} \right)x + \left( {1 - i} \right)y = 1\] and\[2xi + 2y = 1 + i\]. Writing equation in standard form $a + ib$
\[\left( {1 + i} \right)x + \left( {1 - i} \right)y = 1\]
$ \Rightarrow x + ix + y - iy = 1$
$ \Rightarrow \left( {x + y} \right) + i\left( {x - y} \right) = 1$ Or
$ \Rightarrow \left( {x + y} \right) + i\left( {x - y} \right) = 1 + (0)i$
$x + y = 1\,\,and\,\,x - y = 0$

Now, solving two equations $x + y = 1\,\,and\,\,x - y = 0$ obtained in step 1.
On adding two equations we have
$
  2x = 1 \\
   \Rightarrow x = \dfrac{1}{2} \\
 $
Substituting value of $x = \dfrac{1}{2}$ in either of the above equation to get the value of y.
$ \Rightarrow \dfrac{1}{2} + y = 1$
$ \Rightarrow y = 1 - \dfrac{1}{2}$
$ \Rightarrow y = \dfrac{1}{2}$
Therefore, the solution of the first complex equation is$\left( {\dfrac{1}{2},\dfrac{1}{2}} \right)$.

Now, writing second complex equation in standard form $a + ib$
\[2xi + 2y = 1 + i\]Or
\[(2y) + 2xi = 1 + i\]
On comparing real and imaginary part on both sides.
$2y = 1\,\,\,and\,\,2x = 1$
$ \Rightarrow y = \dfrac{1}{2}\,\,and\,\,x = \dfrac{1}{2}$
Therefore, solution of second complex equation is$\left( {\dfrac{1}{2},\dfrac{1}{2}} \right)$
Therefore, from above calculations we see that in step 3 and in step 4 solution obtained of the given complex equation is$\left( {\dfrac{1}{2},\dfrac{1}{2}} \right)$.

Hence, we say that solutions of the given complex equations is one or we can say equations have unique solutions.

Note: In complex problems solution of equations can be obtained by converting equations into standard form.