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Is it \(\large\rm \frac{x+1}{x}=7\) or \(\large\rm x+\frac{1}{x}=7\)
?

(x) + (1/x) = 7, the second one

Yes it is.

\[x+\frac{1}{x}=7\]
\[x=7-\frac{1}{x}\]
\[\frac{1}{x}=7-x\]

Woops typo,\[\large\rm x^2+\frac{1}{x^2}=47\]

\[x^3+\frac{1}{x^2}=(7-\frac{1}{x})^3+\frac{1}{(7-x)^2}\]

\[(x+\frac{1}{x})^2=7^2 \\ (x+\frac{1}{x})^3=7^3\]
some expanding will be involved

seems like there is a more prosaic way of doing this