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Introduction to Planes - Problem 2
So we are still talking about planes in the three dimensional coordinate system. I want to point out that the graph of every equation of this form; ax plus by plus cz equals d, is a plane. So if you are asked to graph something like this, 3x plus y plus 2z is equals to 6. The graph is going to be a plane.
So let me give you an idea of how to graph something like this, really easily. In order to do it, is to plot the intercepts first, the x, y and z intercepts, so let’s do that. Now the x intercept is going to be the point where y and z equals 0. It’s a point on the x axis. And so on the x axis y is 0 and z equals 0. So I substitute 0 for y and z, and I get 3x 0,0 equals 6. So x equals 2. And that means my x intercept is right here. So that’s one point.
Then I’m going to plot my y intercept and I get that by plugging in 0 for x and z. So I get 0 plus y plus 0 equals 6, y equals 6. There it is, 1, 2, 3, 4, 5, 6 right here. So that’s 2 and finally the z intercept, I’m going to plug 0 for x and y and I get 0 plus 0 plus 2z equals 6, so z equals 3. And that’s right here.
And so now, the rest is easy. All I have to do, is draw lines connecting these intercepts. That’s true. I don’t get the entire plane but you never get the whole graph. Even when you are graphing a line and you're just getting a little piece of it.
Here I’m getting a triangular piece of the plane, the triangular piece that is contained in the first octane. And so there it is, a graph of the plane 3x plus y plus 2z, equals 6.