Unique primal optimum implies a unique dual optimum too

Consider the LPP Max $Z=3x+5y$ subject to $x+5y\le10, 2x+2y\le5,\ x,y\ge0$. Which of the following statements is true? $S_1$: There exists a unique optimal solution to the LPP. $S_2$: There exists a unique optimal solution to the dual problem.