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

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Holt Algebra 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

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Holt Algebra Linear Equations in Three Dimensions Graph points and linear equations in three dimensions. Objective

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Holt Algebra Linear Equations in Three Dimensions three-dimensional coordinate system ordered triple z-axis Vocabulary

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Holt Algebra 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 three- dimensional space using a three-dimensional coordinate system, sometimes called coordinate space.

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Holt Algebra 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 an additional coordinate based on the z-axis. Notice that the axes form three planes that intersect at the origin.

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Holt Algebra Linear Equations in Three Dimensions To find an intercept in coordinate space, set the other two coordinates equal to 0. Helpful Hint

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Holt Algebra Linear Equations in Three Dimensions 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)

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Holt Algebra Linear Equations in Three Dimensions 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)

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Holt Algebra Linear Equations in Three Dimensions 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)

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Holt Algebra Linear Equations in Three Dimensions 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

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Holt Algebra Linear Equations in Three Dimensions 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

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Holt Algebra Linear Equations in Three Dimensions Graph the point in three-dimensional space. F(0, 0, 3) From the origin, move 3 units up. y x z F(0, 0, 3) Check It Out! Example 1c

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Holt Algebra Linear Equations in Three Dimensions 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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Holt Algebra Linear Equations in Three Dimensions 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) = –6 x = –3 y-intercept: 2(0) – 3y + (0) = –6 z-intercept: 2(0) – 3(0) + z = –6 y = 2 z = –6

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Holt Algebra Linear Equations in Three Dimensions 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)

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Holt Algebra Linear Equations in Three Dimensions 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 z-intercept: (0) – 4(0) + 2z = 4 y = –1 z = 2 Check It Out! Example 2

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Holt Algebra Linear Equations in Three Dimensions 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

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Holt Algebra Linear Equations in Three Dimensions Track relay teams score 5 points for finishing first, 3 for second, and 1 for third. Lins 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

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Holt Algebra Linear Equations in Three Dimensions Example 3B: Sports Application If Lins 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. Linns team placed first in two events.

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Holt Algebra 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. flashlights 3.50x compasses 1.50y water bottles 0.75z + = = Let x = number of flashlights, y = number of compasses, and z = number of water bottles.

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Holt Algebra 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 z = (6) + 1.5y (24) = y = 15 Use the equation from a. Substitute 6 for x and 24 for z. Solve for y. Steve purchased 15 compasses y + 18 = y = 22.5

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Holt Algebra 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) y x z A( –2, 3, 1) B( 0, –2, 3)

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Holt Algebra Linear Equations in Three Dimensions Lesson Quiz: Part II 3. Graph the linear equation 6x + 3y – 2z = –12 in three-dimensional space.

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Holt Algebra 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. b. If Lily buys 6 pencils and 6 erasers, how many pens can she buy? 0.2x + 0.3y +0.25z = 6 11

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