In linear algebra, why is the graph of a three variable equation of the form $ax+by+cz+d=0$ a plane? With two variables, it is easy to convince oneself that the graph is a line (using similar triangles, for example). However with three variables, this same technique does not seem to work: setting one of the variables to be a constant yields a line in 3-D space (I think), and one could repeat this process for each constant value of that variable, but in the end there seems not to be an easy way to check that all these lines are coplanar.
I don't remember seeing why this is in a book, and Khan Academy's video, for example, simply states that this is the case.