Question 21082
I'm not sure what the "five-step" method is but perhaps you can adapt the following to fit. Let n be the unknown number.

Write the equation:
{{{n + 1/n = 10/3}}} Add the terms on the left side.
Solve the equation:
{{{(n^2 + 1)/n = 10/3}}} Multiply both sides by n.
{{{n^2 + 1 = 10n/3}}} Multiply both sides by 3.
{{{3n^2 + 3 = 10n}}}} Subtract 10n from both sides.
{{{3n^2 - 10n + 3 = 0}}} Solve this quadratic equation for n by factoring.
{{{(3n - 1)(n - 3) = 0}}} Apply the zero products principle.
{{{3n - 1 = 0}}} and/or {{{n - 1 = 0}}}
If {{{3n - 1 = 0}}}, then {{{3n = 1}}} and {{{n = 1/3}}}
If {{{n - 3 = 0}}}, then {{{n = 3}}}

So there are two answers to this problem:
n = 3 and n = 1/3

Check:

n = 3:  {{{3 + 1/3 = 9/3 + 1/3}}} = {{{10/3}}} OK
n = 1/3: {{{1/3 + 1/(1/3) = 1/3 + 3}}} = {{{1/3 + 9/3 = 10/3}}} OK