I need help with the following question.

Explain in general what is the purpose in graphing the TRACE of a plane in 3-D space? Specifically derive each TRACE and use them to draw the graph of 3x+4y+2z=12.

Thanks!!

Your 3-variable equation represents a plane in 3D space.The 'trace' of a plane is a straight line that lies on the plane defined by that equation. To determine a trace, set one of the variables (x)equal to any number (0 is a good one, easier to sketch), then the remaining equation is the trace (a straight line) on the plane at the value of the x variable you have chosen as a reference plane. By repeating this procedure 2 more times, setting another variable (y) equal to any number (say 0), and then the final variable (z) equal to any number (again say 0), you then get three straight lines that will define the plane (they will form a triangle on that plane).

The trace in this example happens to be a straight line, but in general it is a curve that represents the intersection of the surface with the plane given by the value of the variable (x, y, or z) you have chosen to use.

In your example 3x+4y+2z=12, set x=0 , your equation then becomes 4y+2z=12, this is a straight line on your plane through the refernce plane x=0. Then similarly set y =0, get another equation, and z=0 to get the third, and you now have the three lines you can graph to illustrate the given plane. Graphing is not very easy in 3 dimensions, be sure to use 3-D graph paper.