SOLUTION: 1/z = 1/(2 + i) + 1/(-2 + 4i), evaluate z in the form x + iy

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Question 1113676: 1/z = 1/(2 + i) + 1/(-2 + 4i), evaluate z in the form x + iy
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
+1%2Fz+=+%281%2F%282%2Bi%29%29+%2B+%281%2F%28-2%2B4i%29%29+
One way to solve for z:


Multiply both sides by (2+i)(-2+4i), then simplify the RHS:
+%28%28-2%2B4i%29%282%2Bi%29%29%2Fz+=+5i+

Simplify LHS, then multiply both sides by z, then re-write with z on LHS:
+%28z%29%285i%29+=+%28-8%2B6i%29+

Divide both sides by 5i:
++z++=+%28-8%2B6i%29%2F%285i%29+

Multiply RHS by i/i:
+z+=+%28-8i+-+6%29+%2F+-5+

—————
Ans: +highlight%28+z+=+%286%2F5%29%2B%288%2F5%29i++%29+
—————

Another way:
If +1%2Fz+=+1%2Fa+%2B+1%2Fb+ then
+z+=+%28%282%2Bi%29%28-2%2B4i%29%29+%2F+%282%2Bi+-2%2B4i%29++=+%28-8%2B6i%29+%2F+%285i%29+

…and multiplying RHS by i/i: