Confused in how to insert a slack variable in a constrain inequality

According to my understanding , we should put a slack variable to equate an inequality constrain by inserting the slack varaible in the side that is less than the other side , for example if we have 4x+2<2 this will be 4x+2+slack_variable=2 .But here in Wikipedia :
https://en.wikipedia.org/wiki/Slack_variable In the example section , it says the following “By introducing the slack variable y>=0, the inequality Ax<=b can be converted to the equation y-Ax+b = 0” Which means that the slack variable is inserted in the bigger side ! Please some one explain this confusion.

Does the isometry group of a closed simple smooth curve in the plane constrain its perimeter^2/area ratio?

Let $$C$$ be a simple closed smooth curve delimitating a bounded domain $$D$$ in the euclidean plane of isometry group $$G$$ and of given area $$A$$. Does the minimal possible ratio $$\dfrac{P^{2}}{A}$$ where $$P$$ is the perimeter hence the total length of $$C$$ decrease when $$G$$ runs over a sequence $$(G_{i})_{i>0}$$ of groups such that $$i implies $$G_{i}$$ is a strict subgroup of $$G_{j}$$?

Linear programming IFF with equality constrain

Is it possible to write the following logical constrain in linear programming?

Let $$v$$ be an integer variable and $$k$$ an integer constant. Let $$y$$ be a binary variable. The logical constrain requires:

$$y=1 \Longleftrightarrow v=k$$

I need this kind of contrain in linear programming to use it in AMPL, but I really can’t find a way to write it down as a linear constrain.

Thanks to who’ll help me!