𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧: Find the value of x and y for (x+1)/(1+i) + (y-1)/(1-i) = i

𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:

We are given the equation (x+1)/(1+i) + (y-1)/(1-i) = i.

Let's simplify each fraction by multiplying the numerator and denominator by the conjugate of the denominator.

For the first fraction:

(x+1)/(1+i) * (1-i)/(1-i) = (x+1)(1-i) / (1^2 - i^2)

= (x + 1 - xi - i) / (1 - (-1))

= (x + 1 - xi - i) / 2

= (x+1)/2 - (x+1)/2 i

For the second fraction:

(y-1)/(1-i) * (1+i)/(1+i) = (y-1)(1+i) / (1^2 - (-i)^2)

= (y - 1 + yi - i) / (1 - (-1))

= (y - 1 + yi - i) / 2

= (y-1)/2 + (y-1)/2 i

Now substitute these simplified fractions back into the original equation:

((x+1)/2 - (x+1)/2 i) + ((y-1)/2 + (y-1)/2 i) = i

Combine the real and imaginary parts on the left-hand side:

Real part: (x+1)/2 + (y-1)/2 = (x + 1 + y - 1) / 2 = (x + y) / 2

Imaginary part: -(x+1)/2 + (y-1)/2 = (-x - 1 + y - 1) / 2 = (y - x - 2) / 2

So the equation becomes:

(x + y) / 2 + (y - x - 2) / 2 i = 0 + 1i

Now equate the real and imaginary parts:

Real parts:

(x + y) / 2 = 0

x + y = 0 (Equation 1)

Imaginary parts:

(y - x - 2) / 2 = 1

y - x - 2 = 2

y - x = 4 (Equation 2)

Now we have a system of two linear equations with two variables:

1) x + y = 0

2) y - x = 4

From Equation 1, we have y = -x. Substitute this into Equation 2:

(-x) - x = 4

-2x = 4

x = -2

Now substitute the value of x back into Equation 1:

(-2) + y = 0

y = 2

Therefore, the values of x and y are x = -2 and y = 2.