Suppose $x, y \in \mathbb{R}$, and $\mathcal{S_1}$ is a system of inequalities: \begin{align*} \mathcal{S_1} &= \begin{Bmatrix} x - y \geq 1\\ -x + 2y \geq 1\\ 3x - 5y \geq 2 \end{Bmatrix}\\ &= \begin{Bmatrix} x \geq 1 + y\\ x \leq 2y-1\\ x \geq \frac{2+5y}{3} \end{Bmatrix} \end{align*}
I eliminate $x$ from $S_1$ to obtain $S_2$:
\begin{align*} \mathcal{S_2} &= \begin{Bmatrix} 1+y \leq 2y -1 \\ \frac{2+5y}{3} \leq 2y-1 \end{Bmatrix}\\ &= \begin{Bmatrix} 2 \leq y \\ 2+5y \leq 6y-3 \end{Bmatrix}\\ &= \begin{Bmatrix} 2 \leq y \\ 5 \leq y \end{Bmatrix} \end{align*}
Therefore, I know that $5 \leq y < \infty$. Let $S_3$ denote the following system of inequalities
\begin{align*} \mathcal{S_3} &= \begin{Bmatrix} x - y \geq 1\\ -x + 2y \geq 1\\ 3x - 5y \geq 2\\ 5 \leq y < \infty \end{Bmatrix} \end{align*}
My question is, is $S_3 = S_1$?