3 5 Linear Equationsinin Three Dimensions Warm Up

3 -5 Linear. Equationsinin. Three. Dimensions Warm Up Lesson Presentation Lesson Quiz Holt. Mc. Dougal Algebra 2 Holt

3 -5 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 Holt Mc. Dougal Algebra 2 .

3 -5 Linear Equations in Three Dimensions Objective Graph points and linear equations in three dimensions. Holt Mc. Dougal Algebra 2

3 -5 Linear Equations in Three Dimensions Vocabulary three-dimensional coordinate system ordered triple z-axis Holt Mc. Dougal Algebra 2

3 -5 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 threedimensional space using a three-dimensional coordinate system, sometimes called coordinate space. Holt Mc. Dougal Algebra 2

3 -5 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. Holt Mc. Dougal Algebra 2

3 -5 Linear Equations in Three Dimensions Helpful Hint To find an intercept in coordinate space, set the other two coordinates equal to 0. Holt Mc. Dougal Algebra 2

3 -5 Linear Equations in Three Dimensions Example 1 A: 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. Holt Mc. Dougal Algebra 2 z y A(3, – 2, 1) x

3 -5 Linear Equations in Three Dimensions Example 1 B: Graphing Points in Three Dimensions Graph the point in three-dimensional space. B(2, – 1, – 3) z From the origin, move 2 units forward along the x-axis, 1 unit left, and 3 units down. y x B(2, – 1, – 3) Holt Mc. Dougal Algebra 2

3 -5 Linear Equations in Three Dimensions Example 1 C: Graphing Points in Three Dimensions Graph the point in three-dimensional space. C(– 1, 0, 2) z 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. Holt Mc. Dougal Algebra 2 C(– 1, 0, 2) y x

3 -5 Linear Equations in Three Dimensions Check It Out! Example 1 a 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. Holt Mc. Dougal Algebra 2 z y x D(1, 3, – 1)

3 -5 Linear Equations in Three Dimensions Check It Out! Example 1 b Graph the point in three-dimensional space. E(1, – 3, 1) z From the origin, move 1 unit forward along the x -axis, 3 units left, and 1 unit up. E(1, – 3, 1) x Holt Mc. Dougal Algebra 2 y

3 -5 Linear Equations in Three Dimensions Check It Out! Example 1 c Graph the point in three-dimensional space. F(0, 0, 3) z From the origin, move 3 units up. F(0, 0, 3) y x Holt Mc. Dougal Algebra 2

3 -5 Linear Equations in Three Dimensions Recall that the graph of a linear equation in two dimensions is a straight line. In threedimensional 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 Mc. Dougal Algebra 2

3 -5 Linear Equations in Three Dimensions Example 2: Graphing Linear Equations in Three Dimensions Graph the linear equation 2 x – 3 y + z = – 6 in three-dimensional space. Step 1 Find the intercepts: x-intercept: 2 x – 3(0) + (0) = – 6 x = – 3 y-intercept: 2(0) – 3 y + (0) = – 6 y=2 z-intercept: 2(0) – 3(0) + z = – 6 Holt Mc. Dougal Algebra 2

3 -5 Linear Equations in Three Dimensions Example 2 Continued z 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) y (0, 2, 0) x (0, 0, – 6) Holt Mc. Dougal Algebra 2

3 -5 Linear Equations in Three Dimensions Check It Out! Example 2 Graph the linear equation x – 4 y + 2 z = 4 in three-dimensional space. Step 1 Find the intercepts: x-intercept: x – 4(0) + 2(0) = 4 x=4 y-intercept: (0) – 4 y + 2(0) = 4 y = – 1 z-intercept: (0) – 4(0) + 2 z = 4 z=2 Holt Mc. Dougal Algebra 2

3 -5 Linear Equations in Three Dimensions Check It Out! Example 2 Continued z Step 2 Plot the points (4, 0, 0), (0, – 1, 0), and (0, 0, 2). Sketch a plane through the three points. Holt Mc. Dougal Algebra 2 (0, 0, 2) (0, – 1, 0) x ● (4, 0, 0) y

3 -5 Linear Equations in Three Dimensions Example 3 A: 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 first 5 f + Points for second + Points for third = = + + 1 t 3 s Holt Mc. Dougal Algebra 2 30 30 +

3 -5 Linear Equations in Three Dimensions Example 3 B: Sports Application If Lin’s team finishes second in six events and third in two events, in how many events did it finish first? 5 f + 3 s + t = 30 5 f + 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 Mc. Dougal Algebra 2

3 -5 Linear Equations in Three Dimensions Check It Out! Example 3 a 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. flashlights + 3. 50 x + Holt Mc. Dougal Algebra 2 compasses 1. 50 y + + water bottles 0. 75 z = 61. 50

3 -5 Linear Equations in Three Dimensions Check It Out! Example 3 b Steve purchased 6 flashlights and 24 water bottles. How many compasses did he purchase? 3. 5 x + 1. 5 y + 0. 75 z = 61. 50 Use the equation from a. 3. 5(6) + 1. 5 y + 0. 75(24) = 61. 50 Substitute 6 for x and 24 for z. 21 + 1. 5 y + 18 = 61. 50 1. 5 y = 22. 5 y = 15 Steve purchased 15 compasses. Holt Mc. Dougal Algebra 2 Solve for y.

3 -5 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) z B( 0, – 2, 3) A( – 2, 3, 1) y x Holt Mc. Dougal Algebra 2

3 -5 Linear Equations in Three Dimensions Lesson Quiz: Part II 3. Graph the linear equation 6 x + 3 y – 2 z = – 12 in three-dimensional space. Holt Mc. Dougal Algebra 2

3 -5 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. 0. 2 x + 0. 3 y +0. 25 z = 6 b. If Lily buys 6 pencils and 6 erasers, how many pens can she buy? 11 Holt Mc. Dougal Algebra 2