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3-5

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

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  1. 3-5 Linear Equations in Three Dimensions Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

  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

  3. Objective Graph points and linear equations in three dimensions. Vocabulary three-dimensional coordinate system ordered triple z-axis

  4. 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.

  5. 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.

  6. 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) 

  7. 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)

  8. 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.

  9. 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

  10. 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)

  11. 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 +

  12. 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.

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