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

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**3-5**Linear Equations in Three Dimensions Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2**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**Objective**Graph points and linear equations in three dimensions.**Vocabulary**three-dimensional coordinate system ordered triple z-axis**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.**Helpful Hint**To find an intercept in coordinate space, set the other two coordinates equal to 0.**z**y x 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) **z**y x 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)**z**y x 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. **z**y x 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)**z**y x 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) **z**y x 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. **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.**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**z**y x 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. (–3, 0, 0) (0, 2, 0) (0, 0, –6)**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**z**y x 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)**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 third 1t Points for first 5f Points for second 3s 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 Use the equation from A. 5f + 3(6) + (2) = 30 Substitute 6 for s and 2 for t. f = 2 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. 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**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 1.5y = 22.5 Solve for y. y = 15 Steve purchased 15 compasses.**z**y x Lesson Quiz: Part I Graph each point in three dimensional space. 2. B(0, –2, 3) 1. A(–2, 3, 1) B( 0, –2, 3) A( –2, 3, 1) **Lesson Quiz: Part II**3. Graph the linear equation 6x + 3y – 2z = –12 in three-dimensional space.**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? 11

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