Question

x+y/i + x-y+4=0 of value​

Answer 1

SOLUTION

TO DETERMINE

The real values of x and y for which the below is true

$\displaystyle \sf{ \frac{x + y}{i} + x - y + 4 = 0 }$

CONCEPT TO BE IMPLEMENTED

Complex Number :

A complex number z = a + ib is defined as an ordered pair of Real numbers ( a, b) that satisfies the following conditions :

(i) Condition for equality :

(a, b) = (c, d) if and only if a = c, b = d

(ii) Definition of addition :

(a, b) + (c, d) = (a+c, b+ d)

(iii) Definition of multiplication :

(a, b). (c, d) = (ac-bd , ad+bc )

Of the ordered pair (a, b) the first component a is called Real part of z and the second component b is called Imaginary part of z

EVALUATION

Here it is given that

$\displaystyle \sf{ \frac{x + y}{i} + x - y + 4 = 0 }$

$\implies \displaystyle \sf{ \frac{i(x + y)}{ {i}^{2} } + x - y + 4 = 0 }$

$\implies \displaystyle \sf{ \frac{i(x + y)}{ - 1 } + x - y + 4 = 0 }$

$\implies \displaystyle \sf{ - i(x + y) + (x - y + 4 )= 0 }$

Comparing both sides we get

$\sf{x + y = 0 \: \: \: ......(1)}$

$\sf{x - y + 4 = 0 \: \: \: \: .....(2)}$

From Equation (1) we get

$\sf{x = - y}$

From Equation (2) we get

$\sf{x + x + 4 = 0}$

$\implies \sf{2x + 4 = 0}$

$\implies \sf{2x = - 4}$

$\implies \sf{x = - 2}$

From Equation (1) we get

$\sf{y = 2}$

Hence the required values are :

$\sf{x = - 2 \: \: \: \: and \: \: \: \: y = 2}$

FINAL ANSWER

The required values :

$\sf{x = - 2 \: \: \: \: and \: \: \: \: y = 2}$

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