SOLUTION: ab = ½, bc = ⅓, ac = 1/6, find (1/a²) + (1/b²) + (1/c²)

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Question 1209563: ab = ½, bc = ⅓, ac = 1/6,
find (1/a²) + (1/b²) + (1/c²)
Found 2 solutions by Edwin McCravy, greenestamps:
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
system%28ab=1%2F2%2Cbc=1%2F3%2Cac=1%2F6%29

I don't like manipulating fractions, so I'll do it this way to
minimize such manipulation.
 
Multiply the 1st equation by c, the 2nd by a and the 3rd by b

system%28abc=expr%281%2F2%29c%2Cabc=expr%281%2F3%29a%2Cabc=expr%281%2F6%29b%29

So

abc=expr%281%2F2%29c=expr%281%2F3%29a=expr%281%2F6%29b

Multiply through by 6

3c=2a=b

Substitute 2a for b in ab=1/2, then get a, b, and c.

          matrix%283%2C1%2C%0D%0A3c=b%2C%0D%0A3c=%22%22+%2B-+1%2C%0D%0Ac=%22%22+%2B-+1%2F3%29   

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1%5E%22%22%2Fa%5E2+%2B+1%5E%22%22%2Fb%5E2+%2B+1%5E%22%22%2Fc%5E2%22%22=%22%224%2B1%2B9%22%22=%22%2214

Edwin

Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


Note to begin with that the expression we are to evaluate is 1%2Fa%5E2%2B1%2Fb%5E2%2B1%2Fc%5E2, so we don't need to be concerned with positive or negative values of a, b, and c.

Given:
[1] ab=1%2F2
[2] bc=1%2F3
[3] ac=1%2F6

Multiply all three equations together to get

%28abc%29%5E2=1%2F36
[4] abc=1%2F6

From [1] and [4], c=1/3, so 1/c^2=9.
From [2] and [4], a=1/2, so 1/a^2=4.
From [3] and [4], b=1, o 1/b^2=1.

And then

ANSWER: 1%2Fa%5E2%2B1%2Fb%5E2=1%2Fc%5E2=1%2B4%2B9=14