First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightComplex Numberschevron_rightExercise 1.2

Solution

Find the real values of $x$ and $y$ in each part.

(i) $x+iy+2-3i=i(5-i)(3+4i)$

\begin{aligned} & (5-i)(3+4i)=15+20i-3i-4i^2 \\ &=19+17i \\ & i(19+17i)=-17+19i \\ & x+iy+2-3i=-17+19i \\ & (x+2)+i(y-3)=-17+19i \\ & x=-19,\ y=22 \end{aligned}

(ii) $(x+iy)(1-i)=(2-3i)(-5+5i)-\dfrac{3}{5}i$

\begin{aligned} & (2-3i)(-5+5i)=5+25i \\ & 5+25i-\frac{3}{5}i=5+\frac{122}{5}i \\ & (x+iy)(1-i)=(x+y)+i(y-x) \\ & x+y=5,\quad y-x=\frac{122}{5} \\ & y=\frac{147}{10},\ x=-\frac{97}{10} \end{aligned}

(iii) $\dfrac{x}{2+i}+\dfrac{y}{3-i}=4+5i$

\begin{aligned} & \frac{x}{2+i}=\frac{x(2-i)}{5},\quad \frac{y}{3-i}=\frac{y(3+i)}{10} \\ & 2x(2-i)+y(3+i)=40+50i \\ & (4x+3y)+i(-2x+y)=40+50i \\ & 4x+3y=40,\quad -2x+y=50 \\ & x=-11,\ y=28 \end{aligned}