The problem here is that the top and bottom “faces” are annular (a square with a square “hole” removed), and are not simply connected (“like a disk”, as I put it above).

To remedy this, you can add two more edges, joining the inner and outer squares of these two “faces”. This adds no faces or vertices, but increases the number of edges by 2, changing the Euler characteristic to 0 as expected.

As I said in the post, a linear equation (regardless of the number of variables) is one involving polynomials of degree 1, so y = 10 is linear. It is interesting in that it could be either an equation in one variable, y (in this case, one that is already solved), or an equation in two variables, which can be graphed as a horizontal line (because it “doesn’t care” about the value of x, so x can be anything). But either way, it is a linear equation.

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