Holt Algebra 2 3-5 Linear Equations in Three Dimensions 3-5 Linear Equations in Three Dimensions Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson

Holt Algebra 2

3-5 Linear Equations in Three Dimensions 3-5 Linear Equations in Three Dimensions

Holt Algebra 2

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Algebra 2

3-5 Linear Equations in Three Dimensions

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

Holt Algebra 2


Graph points and linear equations in three dimensions.

Objective

Holt Algebra 2


three-dimensional coordinate systemordered triplez-axis

Vocabulary

Holt Algebra 2


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.

Holt Algebra 2


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.

Holt Algebra 2


To find an intercept in coordinate space, set the other two coordinates equal to 0.

Helpful Hint

Holt Algebra 2


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)

Holt Algebra 2



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)

Holt Algebra 2



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)

Holt Algebra 2



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

Holt Algebra 2



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

Holt Algebra 2



F(0, 0, 3)

From the origin, move 3 units up.

y

x

zF(0, 0, 3)

Check It Out! Example 1c

Holt Algebra 2


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.

Holt Algebra 2


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

Holt Algebra 2


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)

Holt Algebra 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) = 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

Holt Algebra 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

Holt Algebra 2


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

Holt Algebra 2


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.

Holt Algebra 2


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.

Holt Algebra 2


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

Holt Algebra 2


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)

Holt Algebra 2


Lesson Quiz: Part II

3. Graph the linear equation 6x + 3y – 2z = –12 in three-dimensional space.

Holt Algebra 2


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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Holt Algebra 2 3-5 Linear Equations in Three Dimensions 3-5 Linear Equations in Three Dimensions Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson