Linear Equations in Three Dimensions

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Linear Equations in Three Dimensions
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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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
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Objective
Graph points and linear equations in three
dimensions.
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Vocabulary
three-dimensional coordinate system ordered
triple z-axis
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A Global Positioning System (GPS) gives locations
using the three coordinates of latitude,
longitude, and elevation. You can represent any
location in three-dimensional space using a
three-dimensional coordinate system, sometimes
called coordinate space.
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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
an additional coordinate based on the z-axis.
Notice that the axes form three planes that
intersect at the origin.
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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.
A(3, 2, 1)
?
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Example 1B Graphing Points in Three Dimensions
Graph the point in three-dimensional space.
B(2, 1, 3)
From the origin, move 2 units forward along the
x-axis, 1 unit left, and 3 units down.
?
B(2, 1, 3)
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Example 1C Graphing Points in Three Dimensions
Graph the point in three-dimensional space.
C(1, 0, 2)
C(1,0, 2)
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.
?
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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.
?
D(1, 3, 1)
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Check It Out! Example 1b
Graph the point in three-dimensional space.
E(1, 3, 1)
From the origin, move 1 unit forward along the
x-axis, 3 units left, and 1 unit up.
E(1, 3, 1)
?
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Check It Out! Example 1c
Graph the point in three-dimensional space.
F(0, 0, 3)
F(0, 0, 3)
From the origin, move 3 units up.
?
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Recall that the graph of a linear equation in two
dimensions is a straight line. In
three-dimensional 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.
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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
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Example 2 Continued
Step 2 Plot the points (3, 0, 0), (0, 2, 0),
and (0, 0, 6). Sketch a plane through the three
points.
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(3, 0, 0)
?
(0, 2, 0)
?
(0, 0, 6)
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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
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Check It Out! Example 2 Continued
(0, 0, 2)
Step 2 Plot the points (4, 0, 0), (0, 1, 0),
and (0, 0, 2). Sketch a plane through the three
points.
?
(0, 1, 0)
?
?
(4, 0, 0)
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Example 3A Sports Application
Track relay teams score 5 points for finishing
first, 3 for second, and 1 for third. Lins 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 third 1t

Points for first 5f
Points for second 3s
30



30

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Example 3B Sports Application
If Lins team finishes second in six events and
third in two events, in how many eventsdid it
finish first?
5f 3s t 30
Use the equation from A.
5f 3(6) (2) 30
Substitute 6 for s and 2 for t.
f 2
Solve for f.
Linns team placed first in two events.
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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.


water bottles 0.75z

flashlights 3.50x
compasses 1.50y
61.50



61.50
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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
Use the equation from a.
3.5(6) 1.5y 0.75(24) 61.50
Substitute 6 for x and 24 for z.
21 1.5y 18 61.50
Solve for y.
1.5y 22.5
y 15
Steve purchased 15 compasses.
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Lesson Quiz Part I
Graph each point in three dimensional space.
1. A(2, 3, 1)
2. B(0, 2, 3)
B( 0, 2, 3)
A( 2, 3, 1)
?
?
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Lesson Quiz Part II
3. Graph the linear equation 6x 3y 2z 12
in three-dimensional space.
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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 pens can she buy?
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