Graphs of Linear Equations in Three Variables

Any equation involving x, y, and z (and possibly higher powers of each) defines a surface in space, not necessarily a flat surface. If the equation is of the general form Ax + By + Cz = D, then its graph is a plane, a flat surface.

In a plane two points determine a line, and in graphing we often choose them to be the x and y intercepts. Similarly, in space any three noncollinear points determine a plane, and if possible we choose these to be the x, y, and z intercepts.

*If all three variables, x, y, and z are present in an equation, the graph will be a triangle in space.

Ex. 2x + 3y + 6z = 12

Using the intercepts we have the points (6,0,0) (0,4,0) (0,0,2). {We will not construct a 3-D box to represent each point.} Locate the 6, 4, and 2 on the respective axes. Draw a line connecting the points.

Notice the equations at each line. If you replace z with 0 in the original equation you get 2x + 3y = 12. Similarly, replacing x and y with 0 will yield the other two equations. The lines from these equations are called traces: a line in which a plane intersects a coordinate plane.

There will not always be three variables present in the original equation.

Ex. Find the x, y, and z intercepts of the graph of 5x + 2z = 10.

Obviously there is no y-intercept. Since there is not a y-intercept that means the graph is a plane parallel to the y-axis. To construct the graph, draw a line connecting the x and z intercepts. From that line, draw a line parallel and then lines parallel to the y-axis.

Ex. 3x + 2z = 0

In this equation, for every point on the y-axis, x = 0 and z = 0. Thus the coordinates of every point on the y-axis satisfy the equation 3x + 2z = 0. This means that the graph of the equation is a plane containing the y-axis.

(4,0,6)

This is the part of the plane containing the y-axis and the point with the coordinates (4,0,6).

**If the constant is not “0”, then the graph is parallel to the axis of the missing variable.

If the constant is “0”, then the graph contains the axis of the missing variable.