Linear equations in 3 d
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Transcript of Linear equations in 3 d
Linear Equations in Three Dimensions
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Warm UpGraph 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
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.
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.
Graph the point in three-dimensional space.
Example 1A: Graphing Points in Three Dimensions
A(3, –2, 1)
From the origin, move 3 units forward along the x-axis, 2 units left, and 1 unit up.
y
x
z
•A(3, –2, 1)
Graph the point in three-dimensional space.
Example 1B: Graphing Points in Three Dimensions
B(2, –1, –3)
From the origin, move 2 units forward along the x-axis, 1 unit left, and 3 units down.
y
x
z
B(2, –1, –3)•
Graph the point in three-dimensional space.
Example 1C: Graphing Points in Three Dimensions
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.
y
x
z
C(–1,0, 2)
•
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.
y
x
z
•D(1, 3, –1)
Check It Out! Example 1a
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.
y
x
z
E(1, –3, 1)•
Check It Out! Example 1b
Graph the point in three-dimensional space.
F(0, 0, 3)
From the origin, move 3 units up.
y
x
zF(0, 0, 3)
•
Check It Out! Example 1c
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.
Graph the linear equation 2x – 3y + z = –6 in three-dimensional space.
Example 2: Graphing Linear Equations in Three Dimensions
Step 1 Find the intercepts:
x-intercept: 2x – 3(0) + (0) = –6x = –3
y-intercept: 2(0) – 3y + (0) = –6
z-intercept: 2(0) – 3(0) + z = –6
y = 2
z = –6
Step 2 Plot the points (–3, 0, 0), (0, 2, 0), and (0, 0, –6). Sketch a plane through the three points.
Example 2 Continued
y
x
z
• (–3, 0, 0)
(0, 2, 0)
(0, 0, –6)•
•
Graph the linear equation x – 4y + 2z = 4 in three-dimensional space.
Step 1 Find the intercepts:
x-intercept: x – 4(0) + 2(0) = 4x = 4
y-intercept: (0) – 4y + 2(0) = 4
z-intercept: (0) – 4(0) + 2z = 4
y = –1
z = 2
Check It Out! Example 2
Step 2 Plot the points (4, 0, 0), (0, –1, 0), and (0, 0, 2). Sketch a plane through the three points.
y
x
z
●(4, 0, 0)
(0, –1, 0)
(0, 0, 2)•
•
Check It Out! Example 2 Continued
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.
Example 3A: Sports Application
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
5f
+++
+
Points for second
3s
Points for third
1t
+ =
=
30
30
Example 3B: Sports Application
If Lin’s team finishes second in six events and third in two events, in how many eventsdid 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.
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.
flashlights
3.50x
+++
compasses
1.50y
water bottles
0.75z
+ =
=
61.50
61.50
Let x = number of flashlights, y = number of compasses, and z = number of water bottles.
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
y = 15
Use the equation from a.
Substitute 6 for x and 24 for z.
Solve for y.
Steve purchased 15 compasses.
21 + 1.5y + 18 = 61.50
1.5y = 22.5
Lesson Quiz: Part I
Graph each point in three dimensional space.
1. A(–2, 3, 1) 2. B(0, –2, 3)
y
x
z
A( –2, 3, 1)•
B( 0, –2, 3)•
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.
b. If Lily buys 6 pencils and 6 erasers, how many pens can she buy?
0.2x + 0.3y +0.25z = 6
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