Answer:
[tex]x=\frac{10}{13}[/tex] and [tex]y=-\frac{24}{13}[/tex]
Step-by-step explanation:
The given complex number equation is:
[tex]\frac{1-2i}{2+i}+\frac{4-i}{3+2i}=x+yi[/tex]
We simplify the LHS and compare with the RHS
We collect LCD on the left to get:
[tex]\frac{(1-2i)(3+2i)+(4-i)(2+i)}{(2+i)(3+2i)}=x+yi[/tex]
[tex]\frac{3+2i-6i+4+8+4i-2i+1}{6+4i+3i-2}=x+yi[/tex]
Simplify to get:
[tex]\frac{16-2i}{4+7i}=x+yi[/tex]
Rationalize the LHS:
[tex]\frac{(16-2i)(4-7i)}{(4+7i)(4-7i)}=x+yi[/tex]
Expand the numerator using the distributive property and the denominator using difference of two squares.
[tex]\frac{64-112i-8i-14}{16+49}=x+yi[/tex]
Simplify to get:
[tex]\frac{50-120i}{65}=x+yi[/tex]
[tex]\frac{10-24i}{13}=x+yi[/tex]
[tex]\frac{10}{13}-\frac{24}{13}i=x+yi[/tex]
By comparing real parts and imaginary parts; we have;
[tex]x=\frac{10}{13}[/tex] and [tex]y=-\frac{24}{13}[/tex]
