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. 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.

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

Example 1A: Graphing Points in Three Dimensions Graph the point in three-dimensional space. A(3, –2, 1) y x z From the origin, move 3 units forward along the x-axis, 2 units left, and 1 unit up. A(3, –2, 1) 

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

Example 1C: Graphing Points in Three Dimensions Graph the point in three-dimensional space. C(–1, 0, 2) y x z 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. 

Graph the point in three-dimensional space. Check It Out! Example 1b Graph the point in three-dimensional space. E(1, –3, 1) y x z From the origin, move 1 unit forward along the x-axis, 3 units left, and 1 unit up. E(1, –3, 1) 

Graph the point in three-dimensional space. Check It Out! Example 1c Graph the point in three-dimensional space. F(0, 0, 3) y x z 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

Example 2 Continued y x 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)  (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

Check It Out! Example 2 Continued y x z (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)

Graph each point in three dimensional space. Lesson Quiz: Part I Graph each point in three dimensional space. 1. A(–2, 3, 1) 2. B(0, –2, 3) y x z 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