Transcript 5.2 Graphing Three Variables

5.2 GRAPHING THREE
VARIABLES
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An equation such as Ax  By  Cz  D such that
A, B, C, and D are not all zero is called a
linear
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
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In space, the graph of a linear equation in three variables is a
plane
.
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
and solve for x.
b. To find the y-intercept, substitute
and solve for y.
c. To find the z-intercept, substitute
and solve for z.

zero
for both y and z
zero
for both x and z
zero
for both x and y
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. Sketch the graph of the equation
5x  2z  10
x-intercept:
y-intercept:
z-intercept:
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What did you find about the graph?
The plane is parallel to the y-axis
If the coefficient of a variable in an equation of a
zero
plane is
, then the plane is
parallel
to the axis of that variable
if the constant term is not zero.
3. Sketch the graph of the equation
3x  2z  0
x-intercept:
y-intercept:
z-intercept:
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What did you find about the graph?
The plane contains the y-axis
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If the coefficient of a variable in an equation of a
zero
plane is
, then the plane
contains
the axis of that variable if
zero
the constant term is
.
4. Sketch the graph of the equation
y3
x-intercept:
y-intercept:
z-intercept:
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What did you find about the graph?
The plane is parallel to the x-axis and z-axis
The plane is parallel to the xz-plane
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two
If the coefficients of
of the variables
are zero, but the constant term is not zero, then the
parallel
graph is
to the other two axes
parallel
and
to the plane of the
other two variables.
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