How do you even do this stuff?!? How do you graph linear equations in three variables using **three dimensional space?**

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The best way to draw these is with isometric paper. If you don't have any, try this:

Draw three line segments meeting at a point so that the angle between them is 120 degrees. Label the segments x,y,z. By convention, the segment towards the top of the paper is z,the segment running towards the right is y, and the one angling to the left is x.

These segments represent the positive x,y, and z axes. You envision each of the axes being perpendicular to each other. One way is to imagine the z and y axes perpendicular in the plane of the paper. (Much as the 2-d Cartesian axes are x and y -- here y is in place of x, z in place of y) Thex-axis is coming out of the paper perpendicular to both.

Extend the segments with dotted lines. Something like this:

The dotted segments represent the negative values along the axes.

(1) To graph (-2,3,1), realize that this is a point. You are usually asked to either draw three arrows (vectors) to get to the point or to draw a "box", with the point as the vertex opposite the origin.

To plot this point, you go back 2 along the x-axis, right 3 parallel to the y-axis, and then up 1 parallel to the z-axis.

(2) The graph of `2x+2y-z=-4` is a plane. The typical way to represent this is with 3 noncollinear points. The easiest points are often the intercepts of the axes. The result is a triangle, but you are supposed to realize that the plane extends indefinitely.

The intercepts are (0,0,4) (Plug in x=0,y=0 and solve for z),(0,-2,0), and (-2,0,0). Plot these points and connect:

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