Mathematics

Simplifying each term by multiplying numerator and denominator by the complex conjugate:
First term: $\frac{5+5i}{3-4i} = \frac{(5+5i)(3+4i)}{3^2 + (-4)^2} = \frac{15 + 20i + 15i + 20i^2}{25} = \frac{15 - 20 + 35i}{25} = \frac{-5 + 35i}{25} = -\frac{1}{5} + \frac{7}{5}i$.
Second term: $\frac{20}{4+3i} = \frac{20(4-3i)}{4^2 + 3^2} = \frac{80 - 60i}{25} = \frac{16}{5} - \frac{12}{5}i$.
Adding the two terms:
$\left(-\frac{1}{5} + \frac{7}{5}i\right) + \left(\frac{16}{5} - \frac{12}{5}i\right) = \frac{-1+16}{5} + \frac{7-12}{5}i = \frac{15}{5} - \frac{5}{5}i = 3 - i$.

Detailed analysis of options:

• $3 - i$ is correct.
• $1 + i$ is incorrect.
• $\frac{5}{2} - \frac{i}{2}$ is incorrect.
• $-2 + 9i$ is incorrect.

Equating real and imaginary components:

• Grouping terms: $(3x + 5y) + i(2y - x) = 7 + 5i$. Equating real parts: $3x + 5y = 7$ (Eq. 1). Equating imaginary parts: $-x + 2y = 5 \implies x = 2y - 5$ (Eq. 2). Substituting Eq. 2 into Eq. 1: $3(2y - 5) + 5y = 7 \implies 6y - 15 + 5y = 7 \implies 11y = 22 \implies y = 2$. Then $x = 2(2) - 5 = 4 - 5 = -1$. Thus $(x, y) = (-1, 2)$.
• Reverses the values of x and y.
• Erroneous solution.
• Reverses the signs.

Multiplication of complex numbers in polar (trigonometric) form:
When multiplying two complex numbers $Z_1 = r_1 e^{i\theta_1}$ and $Z_2 = r_2 e^{i\theta_2}$, their moduli multiply and their arguments add:
$Z_1 Z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)} = r_1 r_2 [\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$.

Option breakdown:

• Angles are multiplied, which is mathematically incorrect.
• Moduli multiply and arguments add, which is the correct polar product identity.
• Arguments are subtracted, which corresponds to division $Z_1 / Z_2$, not multiplication.
• Moduli are divided, which corresponds to division, not multiplication.