Question 1206092
given that z1=2+i and z2=-3+4i and 1/z3=1/z1+1/z2. determine the value of z3 in standard form.

z1=2+i 

 z2=-3+4i

{{{1/z1 = (1/(2+i))*((2-i)/(2-i))= (2-i)/((4-i^2))}}}  = {{{(2-i)/5}}}   ............ (i^2=-i)



{{{1/z2 = (1/(-3+4i))*((-3-4i)/(-3-4i))= (-3-4i)/((9-16i^2))}}}  = {{{(-3-4i)/25}}}   ............ (i^2=-i)
(
{{{ (1/z3) = ( (2-i)/5 )+ ((-3-4i)/25)}}}

Add take LCM

{{{ 1/z3 =  (10-5i +-3-4i)/25}}}

{{{1/z3 =(7-9i)/25}}}

{{{z3= 25/(7-9i)}}}


{{{ z3= (25/(7-9i))*((7+9i)/(7+9i))}}}



{{{ z3 = (25*(7+9i))/(49-81i^2)}}}


{{{ z3 =  (175+225i)/130}}}

{{{ z3= (175/130) +(225/130) i}}}

Simplify to get in standard form

z3=1.346+1.731i