Fourier Motzkin elimination with positive coefficients only How can we use Fourier-Motzkin elimination on system of inequalities with positive coefficients preceding each variable $x_1$ to $x_2$. Obviously, in this case we will only have an upper bound as a solution but how do we find it.
For example:

$5x_1 + 3x_2 \le 8$
$2x_1+5x_2 \le 15$


Rewrite your two inequalities as
$$x_1 \le \frac{8}{5}- \frac{3}{5}x_2 \quad \textrm{and} \quad x_1 \le \frac{15}{2} -\frac{5}{2}x_2$$
Then
$$x_1 \le \min\{\frac{8}{5}- \frac{3}{5}x_2, \frac{15}{2} -\frac{5}{2}x_2 \}$$
This corresponds to the region of $\mathbb{R}^2$ below the graph of the function
$$x_1 = f(x_2) = \left\{ \begin{array}{ll}
\dfrac{8}{5} - \dfrac{3}{5}x_2 & \mbox{if } x_2 \le \dfrac{59}{19} \\
\dfrac{15}{2}-\dfrac{5}{2}x_2 & \mbox{if } x_2 >\dfrac{59}{19} \\
\end{array}\right.$$
