Notion of a "solution" of a (vector/linear system of) equation(s)

[edlugie]

In my linear algebra class today, we were building towards subspaces. Activity 2.3.5
investigates how solution sets of homogeneous vector equations behave under vector operations. I was surprised to see that my students had a lot of trouble parsing the difference between "some vector is the zero vector" versus "the vector solves the equation whose right hand side is zero". I eventually realized this is in part because in their minds they were referring to the RHS of the equation as "the solution". But we want almost the opposite, that the solution is something you plug in (on the left in our conventions) to get the two sides to be equal.

After some discussion, I decided that we should maybe start to call the RHS of a vector equation "the result" to set it apart in our minds. Regardless of if you agree on this term, I think it would be cognitively helpful for students to give some distinct name to the right hand side of these standard form equations.

And after further reflection, I realized that this kind of bumps up against the foundational question "what is an equation?". Is it an operation with a desired result? Is it a claim of equality whose true/false value depends on the values of its variables? Is it just some mix of symbols that makes you "solve for $$x$$"? (Certainly not the last, but I imagine students think this.) I don't know if it's worth the class time to build this into the curriculum, but perhaps this could be in the writing explorations.