Answer

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**Hint:**The problem requires analysis and comparison after solving the Left-hand side of the equation.

**Complete step by step answer:**

The given expression is

$\left( {x + iy} \right)\left( {2 - 3i} \right) = 4 + i......(1)$

The expression consists of a complex number on both sides of the equal to sign and the value of x and y is to be evaluated.

First of all the LHS of the expression should be simplified

LHS

$\left( {x + iy} \right)\left( {2 - 3i} \right)$

Expanding the expression

$

\left( {x + iy} \right)\left( {2 - 3i} \right) \\

2x - 3xi + 2yi - 3y{i^2}......(2) \\

$

The value of ${i^2} = 1$ , substitute it in equation (1)

$2x - 3xi + 2yi + 3y$

Now take the like terms together i.e., terms containing i and variables x and y

$(2x + 3y) + \left( { - 3x + 2y} \right)i$

Now the equation (1) becomes as

$(2x + 3y) + \left( { - 3x + 2y} \right)i = 4 + i......(3)$

On comparing the two sides of the equation (3),

$2x + 3y = 4......(i)$

$ - 3x + 2y = 1......(ii)$

In comparison, two linear equations come out in the form of x and y. Solving equation(i) and equation (ii) the values of x and y can be evaluated.

Multiply equation (i) by 3 and equation (ii) by 2,

$6x + 9y = 12......(iii)$

$ - 6x + 4y = 2......(iv)$

On adding equation (iii) and (iv) , the value of y can be calculated.

\[

\underline

6x + 9y = 12 \\

- 6x + 4y = 2 \\

\\

0 + 13y = 14 \\

y = \dfrac{{14}}{{13}} \\

\]

The value of y comes out to be $y = \dfrac{{14}}{5}$ , substitute it in equation (i) to calculate the value of x.

$

2x + 3\left( {\dfrac{{14}}{{13}}} \right) = 4 \\

2x + \dfrac{{42}}{{13}} = 4 \\

2x = 4 - \dfrac{{42}}{{13}} \\

2x = \dfrac{{52 - 42}}{{13}} \\

2x = \dfrac{{10}}{{13}} \\

x = \dfrac{{10}}{{2 \times 13}} \\

x = \dfrac{5}{{13}} \\

$

The value comes out to be $x = \dfrac{5}{{13}}$

Thus, $\left( {x,y} \right) = \left( {\dfrac{5}{{13}},\dfrac{{14}}{{13}}} \right)$

Hence, option (C) is correct.

**Note:**

The important step in the evaluation of this problem is solving the LHS part and it’s a comparison that should be kept in mind.

A complex number is a number that has a real and imaginary part . for instance, is a complex number in which x is the real part, and y is the imaginary part.

The value of and the value of. This should be kept in mind while evaluating the equality involving the complex numbers.

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