Verify if x=3/2 and y=-1/2 satisfy the equation (2+j)/(1−j) = j(x+jy).

To simplify the fraction, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of $1-j$ is $1+j$.

$$\frac{2+j}{1-j} = \frac{2+j}{1-j} \times \frac{1+j}{1+j}$$

Multiply the numerators:

$$(2+j)(1+j) = 2(1) + 2(j) + j(1) + j(j)$$ $$= 2 + 2j + j + j^2$$

Since $j^2 = -1$:

$$= 2 + 3j - 1 = 1 + 3j$$

Multiply the denominators:

$$(1-j)(1+j) = 1^2 - j^2 = 1 - (-1) = 1 + 1 = 2$$

So, the LHS simplifies to:

$$\frac{1+3j}{2} = \frac{1}{2} + j\frac{3}{2}$$

$$j(x+jy) = jx + j^2y$$

Since $j^2 = -1$:

$$= jx - y = -y + jx$$

$$\frac{1}{2} + j\frac{3}{2} = -y + jx$$

For two complex numbers to be equal, their real parts must be equal, and their imaginary parts must be equal.
Equating the real parts:

$$\frac{1}{2} = -y$$ $$y = -\frac{1}{2}$$

Equating the imaginary parts:

$$\frac{3}{2} = x$$ $$x = \frac{3}{2}$$

The calculated values are $x=\frac{3}{2}$ and $y=-\frac{1}{2}$. These match the proposed values.

✅ Correct. The values $x=\frac{3}{2}$ and $y=-\frac{1}{2}$ are correct because when substituted into the equation, they satisfy the equality of the complex numbers.