The answer is yes, and in fact one can even reduce to the decision problem of linear inequalities feasibility!
We are as input given a LP instance P:
$\max c^Tx\ \text{s.t.}\ Ax \leq b\ ;\ x\geq 0$.
We furthermore have access to an oracle that given a system of inequalities $S=\{Bz \leq d\}$ returns yes/no, whether the system is feasible.
The reduction now proceeds as follows:
- Test if $S_1=\{Ax \leq b\ ;\ x \geq 0\}$ is feasible. If not, we can report that P is INFEASIBLE.
- Form the dual program D: $\min b^Ty\ \text{s.t.}\ A^Ty \geq c\ ;\ y \geq 0$.
- Test if $S_2=\{Ax \leq b\ ;\ x\geq 0\ ;\ A^Ty\geq c\ ;\ y\geq0 \ ;\ b^Ty \leq c^Tx\}$ is feasible. If not, we can report that P is UNBOUNDED.
- Iterate over the inequalities of $S_1$ and try to add them one-by-one as equalities (i.e. add the reverse inequality) to the system $S_2$. If the system remains feasible we keep the constraint in $S_2$, and otherwise remove it again. Let $S_3$ be the system of constraints (linear equalities) that gets added in this way. The system $S_3$ will now completely determine an optimal basic solution to P.
- Using Gaussian Elimination on the system $S_3$ compute an optimal solution $x$ to P.