### 3.5 Graphs in Three Dimensions 3

```3-5
3-5 Linear
LinearEquations
EquationsininThree
ThreeDimensions
Dimensions
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Holt
Algebra
Algebra
22
3-5 Linear Equations in Three Dimensions
Warm Up
Graph each of the following points in the
coordinate plane.
1. A(2, –1)
2. B(–4, 2)
3. Find the intercepts of the line
x: –9; y: 3
Holt Algebra 2
.
3-5 Linear Equations in Three Dimensions
Objective
Graph points and linear equations in three
dimensions.
Holt Algebra 2
3-5 Linear Equations in Three Dimensions
Vocabulary
three-dimensional coordinate system
ordered triple
z-axis
Holt Algebra 2
3-5 Linear Equations in Three Dimensions
A Global Positioning
System (GPS) gives
locations using the three
coordinates of latitude,
longitude, and elevation.
You can represent any
location in threedimensional space using
a three-dimensional
coordinate system,
sometimes called
coordinate space.
Holt Algebra 2
3-5 Linear Equations in Three Dimensions
Each point in coordinate
space can be represented
by an ordered triple of
the form (x, y, z). The
system is similar to the
coordinate plane but has
based on the z-axis.
Notice that the axes form
three planes that intersect
at the origin.
Holt Algebra 2
3-5 Linear Equations in Three Dimensions
To find an intercept in coordinate space, set the
other two coordinates equal to 0.
Holt Algebra 2
3-5 Linear Equations in Three Dimensions
Example 1A: Graphing Points in Three Dimensions
Graph the point in three-dimensional space.
A(3, –2, 1)
From the origin,
move 3 units
forward along the
x-axis, 2 units left,
and 1 unit up.
Holt Algebra 2
z
y
A(3, –2, 1)

x
3-5 Linear Equations in Three Dimensions
Example 1B: Graphing Points in Three Dimensions
Graph the point in three-dimensional space.
B(2, –1, –3)
z
From the origin,
move 2 units
forward along the
x-axis, 1 unit left,
and 3 units down.
y
x

B(2, –1, –3)
Holt Algebra 2
3-5 Linear Equations in Three Dimensions
Example 1C: Graphing Points in Three Dimensions
Graph the point in three-dimensional space.
C(–1, 0, 2)
z
From the origin,
move 1 unit back
along the x-axis, 2
units up. Notice that
this point lies in the
xz-plane because the
y-coordinate is 0.
Holt Algebra 2
C(–1,0, 2)

y
x
3-5 Linear Equations in Three Dimensions
Check It Out! Example 1a
Graph the point in three-dimensional space.
D(1, 3, –1)
From the origin,
move 1 unit forward
along the x-axis, 3
units right, and 1 unit
down.
Holt Algebra 2
z
y

x
D(1, 3, –1)
3-5 Linear Equations in Three Dimensions
Check It Out! Example 1b
Graph the point in three-dimensional space.
E(1, –3, 1)
z
From the origin, move
1 unit forward along
the x-axis, 3 units left,
and 1 unit up.
E(1, –3, 1)

x
Holt Algebra 2
y
3-5 Linear Equations in Three Dimensions
Check It Out! Example 1c
Graph the point in three-dimensional space.
F(0, 0, 3)
z
From the origin, move
3 units up.

F(0, 0, 3)
y
x
Holt Algebra 2
3-5 Linear Equations in Three Dimensions
Recall that the graph of a linear equation in two
dimensions is a straight line. In threedimensional space, the graph of a linear
equation is a plane. Because a plane is defined
by three points, you can graph linear equations
in three dimensions by finding the three
intercepts.
Holt Algebra 2
3-5 Linear Equations in Three Dimensions
Example 2: Graphing Linear Equations in Three
Dimensions
Graph the linear equation 2x – 3y + z = –6 in
three-dimensional space.
Step 1 Find the intercepts:
x-intercept: 2x – 3(0) + (0) = –6
x = –3
y-intercept: 2(0) – 3y + (0) = –6
y=2
z-intercept: 2(0) – 3(0) + z = –6
z = –6
Holt Algebra 2
3-5 Linear Equations in Three Dimensions
Example 2 Continued
z
Step 2 Plot the
points (–3, 0, 0),
(0, 2, 0), and
(0, 0, –6). Sketch a
plane through the
three points.
 (–3, 0, 0)

y
(0, 2, 0)
x
 (0, 0, –6)
Holt Algebra 2
3-5 Linear Equations in Three Dimensions
Check It Out! Example 2
Graph the linear equation x – 4y + 2z = 4 in
three-dimensional space.
Step 1 Find the intercepts:
x-intercept: x – 4(0) + 2(0) = 4
x=4
y-intercept: (0) – 4y + 2(0) = 4
y = –1
z-intercept: (0) – 4(0) + 2z = 4
z=2
Holt Algebra 2
3-5 Linear Equations in Three Dimensions
Check It Out! Example 2 Continued
z
Step 2 Plot the
points (4, 0, 0),
(0, –1, 0), and
(0, 0, 2). Sketch a
plane through the
three points.
Holt Algebra 2
(0, 0, 2)

(0, –1, 0)

x
●
(4, 0, 0)
y
3-5 Linear Equations in Three Dimensions
Example 3A: Sports Application
Track relay teams score 5 points for finishing
first, 3 for second, and 1 for third. Lin’s team
scored a total of 30 points.
Write a linear equation in three variables to
represent this situation.
Let f = number of races finished first, s = number of
races finished second, and t = number of races
finished third.
Points for first + Points for second
5f
Holt Algebra 2
+
3s
+ Points for third =
=
+
1t
30
30
+
3-5 Linear Equations in Three Dimensions
Example 3B: Sports Application
If Lin’s team finishes second in six events and
third in two events, in how many events
did it finish first?
5f + 3s + t = 30
5f + 3(6) + (2) = 30
f=2
Use the equation from A.
Substitute 6 for s and 2 for t.
Solve for f.
Linn’s team placed first in two events.
Holt Algebra 2
3-5 Linear Equations in Three Dimensions
Check It Out! Example 3a
Steve purchased \$61.50 worth of supplies for
a hiking trip. The supplies included flashlights
for \$3.50 each, compasses for \$1.50 each, and
water bottles for \$0.75 each.
Write a linear equation in three variables to
represent this situation.
Let x = number of flashlights, y = number of
compasses, and z = number of water bottles.
flashlights +
3.50x
Holt Algebra 2
+
compasses
1.50y
+
+
water bottles
0.75z
=
61.50
=
61.50
3-5 Linear Equations in Three Dimensions
Check It Out! Example 3b
Steve purchased 6 flashlights and 24 water
bottles. How many compasses did he purchase?
3.5x + 1.5y + 0.75z = 61.50
3.5(6) + 1.5y + 0.75(24) = 61.50
21 + 1.5y + 18 = 61.50
1.5y = 22.5
y = 15
Steve purchased 15 compasses.
Holt Algebra 2
Use the equation
from a.
Substitute 6 for x
and 24 for z.
Solve for y.
3-5 Linear Equations in Three Dimensions
Lesson Quiz: Part I
Graph each point in three dimensional space.
1. A(–2, 3, 1)
2. B(0, –2, 3)
z
B( 0, –2, 3)

A( –2, 3, 1)

y
x
Holt Algebra 2
3-5 Linear Equations in Three Dimensions
Lesson Quiz: Part II
3. Graph the linear equation 6x + 3y – 2z = –12 in
three-dimensional space.
Holt Algebra 2
3-5 Linear Equations in Three Dimensions
Lesson Quiz: Part III
4. Lily has \$6.00 for school supplies. Pencils cost
\$0.20 each, pens cost \$0.30 each, and erasers
cost \$0.25 each.
a. Write a linear equation in three variables to
represent this situation.
0.2x + 0.3y +0.25z = 6
b. If Lily buys 6 pencils and 6 erasers, how many