1. Graphing Linear Equations in Three Variables
GOAL Graph linear equations in three variables.
Key Words
• three-dimensional
coordinate system
• z-axis
• ordered triple
• octants
• linear equation in three
variables

2. Points in a two-dimensional coordinate system are represented by ordered pairs. Each point in
space can be represented by an ordered triple (x, y, z), such as (1, -2, 3).
The three axes, taken two at a time, determine three coordinate planes. These planes divide space into eight octants. The first octant is the one for which all three coordinates are positive.

3. Plot the ordered triple in a three-dimensional coordinate system.
Example 1
Plot Points in Three Dimensions
Plot the ordered triple in a three-dimensional coordinate system. a.
5, 3, 4 ( ) – b.
3, 4, 2 ( ) – a.
5, 3, 4 ( ), –
5, 3 )
SOLUTION
To plot first find the point in the xy-plane. The point lies 4 units above it.

4. Example 1
Plot Points in Three Dimensions b.
To plot find the point in the xy-plane. The point lies 2 units below it.
3, 4, 2 ( ) –
3, 4 4

5. Plot the ordered triple in a three-dimensional coordinate system.
Checkpoint
Plot Points in Three Dimensions
Plot the ordered triple in a three-dimensional coordinate system. 1.
1, 2, 4 ( )
ANSWER

6. Plot the ordered triple in a three-dimensional coordinate system.
Checkpoint
Plot Points in Three Dimensions
Plot the ordered triple in a three-dimensional coordinate system. 2.
2, 1, 3 ( ) –
ANSWER

7. Plot the ordered triple in a three-dimensional coordinate system.
Checkpoint
Plot Points in Three Dimensions
Plot the ordered triple in a three-dimensional coordinate system. 3. – – – (
4, 1, 5 )
ANSWER

8. Plot the ordered triple in a three-dimensional coordinate system.
Checkpoint
Plot Points in Three Dimensions
Plot the ordered triple in a three-dimensional coordinate system. 4.
3, 2, 4 ( ) –
ANSWER

9. The graph of a linear equation in three variables is a plane.
A linear equation in three variables x, y, and z is an equation of the form
ax + by + cz = d
where the constants a, b, and c are not all zero.
An ordered triple (x, y, z) is a solution of a linear equation in three variables if the values of x, y, and z make the equation true.
The graph of a linear equation in three variables is a plane.
In a three-dimensional coordinate system, the graphs of equations in one or two variables are also planes.

11. Find the points where the graph intersects the axes.
Example 2
Graph a Linear Equation in Three Variables
Sketch the graph of 3x 2y 4z Label the points where the graph crosses the x-, y-, and z-axes. + =
Find the points where the graph intersects the axes.
First let x and y 0, and then solve for z.
SOLUTION =
Substitute 0 for x and for y. 12 = + 4z 3 ( ) 2
Simplify. 12 = 4z 3 = z
Solve for z.
The z-intercept is 3, so plot the point ( )
0, 0, 3

12. Next, let y 0 and z 0, and then solve for x. = =
Example 2
Graph a Linear Equation in Three Variables
Next, let y and z 0, and then solve for x. = =
Substitute 0 for y and for z. 12 = + 3x 2 ( ) 4
Simplify. 12 = 3x 4 = x
Solve for x.
The x-intercept is 4, so plot the point
Finally, let x and z 0, and then solve for y. = ( )
4, 0, 0
Substitute 0 for x and for z. 12 = + 2y 3 ( ) 4
Simplify. 12 = 2y 6 = y
Solve for y.
The y-intercept is 6, so plot ( )
0, 6, 0

13. Example 2
Graph a Linear Equation in Three Variables
Connect the points with lines. The lines form the triangular region of the plane that lies in the first octant.

14. Checkpoint
Graph a Linear Equation in Three Variables
Sketch the graph of the equation. Label the points where the graph crosses the x-, y-, and z-axes. 5. 6 = + x 3y 2z
ANSWER

15. Checkpoint
Graph a Linear Equation in Three Variables
Sketch the graph of the equation. Label the points where the graph crosses the x-, y-, and z-axes. 6. 10 = + y 5x 2z
ANSWER

16. Checkpoint
Graph a Linear Equation in Three Variables
Sketch the graph of the equation. Label the points where the graph crosses the x-, y-, and z-axes. 7. 4 = y + x
ANSWER

17. GARDENING Tulips come in a variety of sizes, shapes, and colors
GARDENING Tulips come in a variety of sizes, shapes, and colors. The bulbs must be planted in late fall or early winter in order to bloom the
following spring.

18. Write a linear model for the total cost of the bulbs
Example 3
Model a Real-Life Situation
Gardening You are planting tulip bulbs and daffodil bulbs in a community garden. Tulip bulbs cost $6 per dozen, and daffodil bulbs cost $5 per dozen. Fertilizer for the bulbs costs $10.
Write a linear model for the total cost of the bulbs
and the fertilizer for the garden.
b. Evaluate the model for several different numbers of tulip bulbs and daffodil bulbs. Organize your results in a table. 18

19. Your total cost involves two variable costs (for the
Example 3
Model a Real-Life Situation
SOLUTION
Your total cost involves two variable costs (for the
two types of bulbs) and one fixed cost (for the
fertilizer).
VERBAL
MODEL
Total
cost • =
Dozens of
tulip
bulbs
Cost of
(per
dozen) +
daffodil
Cost
fertilizer 19

20. Cost of tulip bulbs (per dozen) 6 (dollars) =
Example 3
Model a Real-Life Situation
LABELS
Total cost C
(dollars) =
Cost of tulip bulbs (per dozen) 6
(dollars) =
Dozens of tulip bulbs t
(dozens of bulbs) =
Cost of daffodil bulbs (per dozen) 5
(dollars) =
Dozens of daffodil bulbs d
(dozens of bulbs) =
Cost of fertilizer
(dollars) =
ALGEBRAIC
MODEL = C 6t + 5d 10 20

21. t (dozens of tulip bulbs)
Example 3
Model a Real-Life Situation
b. To evaluate the model, substitute values for t and d. For example, substitute 4 for t and 3 for d. = C 6t + 5d 10
Write algebraic model. = C 6 + 5
Substitute 4 for t and 3 for d. ( ) 4 3 10 = C 49
Simplify.
The table shows the total
cost for several more
values of t and d.
t (dozens of tulip bulbs)
daffodil bulbs)
d (dozens of 2 4 6 1 3
$27
$32
$37
$39
$44
$49
$51
$56
$61
The total cost is $49 for 4
dozen tulip bulbs and 3
dozen daffodil bulbs.

22. Use the given coordinates to find the coordinates of
the vertices J, K, L, and M of the rectangular prism.

23. Use the given point to find the volume of the rectangular prism.

24. Geometry Skills: Parallel Lines and Angles
Find the values of x and y.

29. Slip

30. Homework