How to find all the positive integral solutions of $5x+7y=100$? How to find the number of all the positive integral solutions and the solutions itselves of $$5x+7y=100?$$

Please help me!!
Assume we have $5x+7y=100$. Let us solve for the variable with the smallest coefficient $x=20-\frac{7y}{5}$. This means that $\frac{7y}{5}$ is integer. Therefore $y=5y_2$ for some integer $y_2$. 

Plugging this into the original equation we get $x=20-7y_2$. You see now, that for every integer you put in $y_2$ you get an integer value for $x$, and an integer value for $y=5y_2$ that satisfy the equation $5x+7y=100$.

Since we want positive solutions you can put $y_2&gt;0$ such that $20-7y_2&gt;0$, i.e. $20/7&gt;y_2$. So, $0&lt;y_2\leq 2$. 

We have then two possibilities: $y_2=1$ or $y_2=2$, for which we get $y=5$, or $y=10$ respectively. The corresponding values for $x$ are $x=13$, or $6$.
Thank you for pointing out my mistake, postmortem. 
Through diophantine equation, I get 

$$x = 300 -7r,$$ 
for all $r-$ integers, 
$$y = -200 +5r $$

To solve the inequalities above. So, to get the positive solution, for the above inequalities, 

we need to set certain boundaries such that, 
$r\ge40$ , and $r&lt;42$, 

sorry for my mistake, due to I am having fever. Thank you so much for pointing out my mistake. 

Reference：
How to find solutions of linear Diophantine ax + by = c?