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

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson  -> Lesson -> SOLUTION: given that z1=2+i and z2=-3+4i and 1/z3=1/z1+1/z2. determine the value of z3 in standard form.       Log On


   

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.
Found 3 solutions by mananth, Edwin McCravy, math_tutor2020:
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
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
= %282-i%29%2F5 ............ (i^2=-i)


= %28-3-4i%29%2F25 ............ (i^2=-i)
(
+%281%2Fz3%29+=+%28+%282-i%29%2F5+%29%2B+%28%28-3-4i%29%2F25%29
Add take LCM
+1%2Fz3+=++%2810-5i+%2B-3-4i%29%2F25
1%2Fz3+=%287-9i%29%2F25
z3=+25%2F%287-9i%29

+z3=+%2825%2F%287-9i%29%29%2A%28%287%2B9i%29%2F%287%2B9i%29%29


+z3+=+%2825%2A%287%2B9i%29%29%2F%2849-81i%5E2%29

+z3+=++%28175%2B225i%29%2F130
+z3=+%28175%2F130%29+%2B%28225%2F130%29+i
Simplify to get in standard form
z3=1.346+1.731i










Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
The other tutor got you here:

+z%5B3%5D=+expr%28175%2F130%29+%2Bexpr%28225%2F130%29+i

But don't go to rounded off decimals, 
as they aren't exact answers.

Reduce the fractions instead:

+z%5B3%5D=+expr%2835%2F26%29+%2Bexpr%2845%2F26%29+i

Then if you like, you can factor out 5%2F26

z%5B3%5D=expr%285%2F26%29%287%2B9i%29

Edwin

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Answer:
z%5B3%5D+=+35%2F26%2Bexpr%2845%2F26%29i


Work Shown

1%2F%28z%5B3%5D%29+=+1%2F%28z%5B1%5D%29%2B1%2F%28z%5B2%5D%29

Get every fraction on the right hand side in terms of the LCD

1%2F%28z%5B3%5D%29+=+%28z%5B2%5D%2Bz%5B1%5D%29%2F%28z%5B1%5Dz%5B2%5D%29

z%5B3%5D+=+%28z%5B1%5Dz%5B2%5D%29%2F%28z%5B2%5D%2Bz%5B1%5D%29 Apply reciprocal to both sides.

z%5B3%5D+=+%28%282%2Bi%29%28-3%2B4i%29%29%2F%28-3%2B4i%2B2%2Bi%29

z%5B3%5D+=+%28-10%2B5i%29%2F%28-1%2B5i%29 See scratch work section shown below.

z%5B3%5D+=+%28%28-10%2B5i%29%28-1-5i%29%29%2F%28%28-1%2B5i%29%28-1-5i%29%29 Multiply top and bottom by conjugate of the denominator so the denominator becomes a real number.

z%5B3%5D+=+%2835%2B45i%29%2F%2826%29 Follow similar steps as the scratch work section shown below. Note: (a+bi)(a-bi) = a^2+b^2

z%5B3%5D+=+35%2F26%2Bexpr%2845%2F26%29i

Complex number z%5B3%5D is of the form a+bi where a+=+35%2F26 and b+=+45%2F26

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Scratch Work

%282%2Bi%29%28-3%2B4i%29+=+2%28-3%2B4i%29%2Bi%28-3%2B4i%29

%282%2Bi%29%28-3%2B4i%29+=+-6%2B8i-3i%2B4i%5E2

%282%2Bi%29%28-3%2B4i%29+=+-6%2B8i-3i%2B4%28-1%29

%282%2Bi%29%28-3%2B4i%29+=+-6%2B8i-3i-4

%282%2Bi%29%28-3%2B4i%29+=+-10%2B5i
Another approach for this scratch work section is to use the FOIL rule or the box method.