An equation in two variables represents a line in a plane. An equation in three variables represents a plane in space.

Activity 2

Given the following equation in three variables, draw the plane in a coordinate space. x + 2 y − z = 6 x plus 2 y minus z equals 6

7. Let x = 0. Graph the resulting equation in the yz plane.
8. Let y = 0. Graph the resulting equation in the xz plane.

From geometry you know that two lines determine a plane.

9. Sketch the plane x + 2 y − z = 6 . x plus 2 y minus z equals 6 .

   (If you need help, find a third line by letting z = 0 and then graph the resulting equation in the xy plane.)

Activity 3

Two equations in three variables represent two planes in space.

10. Draw the two planes determined by the following equations:

    2 x + 3 y − z = 12 2 x − 4 y + z = 8 table with 2 rows and 1 column , row1 column 1 , 2 x plus 3 y minus z equals 12 , row2 column 1 , 2 x minus 4 y plus z equals 8 , end table

11. Describe the intersection of the two planes above.

Exercises

Find the coordinates of each point in the diagram.

Image Long Description

12. A
13. B
14. C
15. D
16. E
17. F

Sketch the graph of each equation.

18. x − y − 4 z = 8 x minus , y minus 4 z equals 8
19. x + y + z = 2
20. − 3 x + 5 y + 10 z = 15 negative 3 x plus 5 y plus 10 z equals 15
21. 6 x + 6 y − 12 z = 36 6 x plus 6 y minus 12 z equals 36

Graph the following pairs of equations in the same coordinate space and describe their intersection, if any.

22. − x + 3 y + z = 6 − 3 x + 5 y − 2 z = 60 table with 2 rows and 1 column , row1 column 1 , negative x plus 3 y plus z equals 6 , row2 column 1 , negative 3 x plus 5 y minus 2 z equals 60 , end table
23. − 2 x − 3 y + 5 z = 7 2 x − 3 y − 4 z = − 4 table with 2 rows and 1 column , row1 column 1 , negative 2 x minus 3 y plus 5 z equals 7 , row2 column 1 , 2 x minus 3 y minus 4 z equals negative 4 , end table

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