5.2 Graphing Three Variables
An equation such as Ax  By  Cz  D such that A, B, C, and D are not all zero is called a
equation in three variables.
We can regard an equation in one or two variables as an equation in three variables.
Example: 4x  3z  5 can be thought of as 4x  0y  3z  5
In space, the graph of a linear equation in three variables is a
.
In order to graph a linear equation in three variables we find the x-, y-, and z-intercepts.
a. To find the x-intercept, substitute
for both y and z and solve for x.
b. To find the y-intercept, substitute
for both x and z and solve for y.
c. To find the z-intercept, substitute
for both x and y and solve for z.
We can then plot the intercepts and graph the plane that goes through them.
1. Sketch the graph of the equation 2x  3y  6z  12
x-intercept:
y-intercept:
z-intercept:
2. Find the x-, y-, and z-intercepts and graph 5x  2z  10 .
x-intercept:
y-intercept:
z-intercept:
What did you find about the graph?
If the coefficient of a variable in an equation of a plane is
, then the plane is
to the axis of that variable if the constant term is not zero.
3. Find the x-, y-, and z-intercepts and graph 3x  2z  0 .
x-intercept:
y-intercept:
z-intercept:
What did you find about the graph?
If the coefficient of a variable in an equation of a plane is
the axis of that variable if the constant term is
, then the plane
.
4. Find the x-, y-, and z-intercepts and graph y  3 .
x-intercept:
y-intercept:
z-intercept:
What did you find about the graph?
If the coefficients of
graph is
variables.
of the variables are zero, but the constant term is not zero, then the
to the other two axes and parallel to the plane of the other two
Questions 5-8: A. Name the x-, y-, and z-intercepts. B. Determine whether or not the graph is parallel to or
contains any of the coordinate axes, and if so, which one(s). C. Determine whether or not the graph is
parallel to or coincides with one of the coordinate planes, and if so, which one. D. Graph.
5.
6x  3y  2z  18
6.
3y  2z  6
7.
x  2
8.
3x  5y  3