Prove that 1/x + 1/y + 1/z = 0.

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If 2x = 3y = 6-z, prove that 1/x + 1/y + 1/z = 0.

Solution:

Consider

2x = 3y = 6-z = k

Here

2x = k

We can write it as

2 = (k)1/x

3y = k

We can write it as

3 = (k)1/y

6-z = k

We can write it as

6 = (k)-1/z

So we get

2 × 3 = 6

(k)1/x × (k)1/y = (k)-1/z

By further calculation

(k)1/x + 1/y = (k)-1/z

We get

1/x + 1/y = – 1/z

1/x + 1/y + 1/z = 0

Therefore, it is proved.

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