= Solution
Take the dual vector $y=(8/17,45/17)^T$. Its column constraints have values $5,12,53/17,233/17$, exceeding or equaling $c=(5,12,2,5)^T$; its entries are positive. Thus it is dual feasible. The first two constraints are equalities while the last two have strictly positive slack, suggesting $x_3=x_4=0$. Solve the two primal equalities to get $x_1=5/17$, $x_2=3/17$, both positive.
The resulting feasible objectives agree:
$$
\boxed{x_*=(5/17,3/17,0,0)^T,\quad y_*=(8/17,45/17)^T,\quad\text{optimal value }61/17.}
$$
<Weak duality> proves optimality. Strict dual slack in columns three and four and positive dual entries force the two primal equalities, proving primal uniqueness; positive first two primal coordinates force the two <independent> dual equalities, proving dual uniqueness.
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