So each of three formulas has three variables. I am trying to picture this geometrically as if it were position variables in 3D space. Then each formula describes a unique point in 3D space, and the row vector goes from the 000 point to the point described by the formula. So, there are three points, a triangle, sitting in 3D space, each of the three vertices forming a vector from the point of origin. So I begin to use fundamental row operations to get the matrix in row echelon form. What happens to my geometric triangle? Substituting one row for another just changes the axis labels....multiplying one row by a scaler makes the row vector longer. Adding two row vectors and substituting the result back into one of the rows...I am having a bit of difficulty making a picture of that. Then, I solve the matrix for the three variables algebraically, and get another end point to make a vector with the origin. I am thinking of each vector as a push on an object....the push of the solution vector does the same thing to the object as the three row vectors would, if it replaced them. So, how do I get from flipping triangles to the meaning of the algebraic solution? My first guess is that someone may tell me I am going about it all wrong. I don't like to memorize math manipulations without knowing what they do, geometrically. Surely this should work, at least in 3D? Any help appreciated, thanks.