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What you will learn
Vocabulary How to plot a point in 3 dimensional
space How to plot a plane in 3 dimensional
space How to solve a system of equations
when you have three variables
Objective: 3.5 and 3.6 Graphing and Solving in 3D
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Graphing in 3 Dimensions
Solving equations in three variables can be pictured using a three-dimensional coordinate system.
In addition to the x and y axis, we add the z axis.
z Z+
Z-X+
X-
Y+
Y-
Objective: 3.5 and 3.6 Graphing and Solving in 3D
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More Vocabulary
Each point in space can be represented by an ordered triple (x, y, z).
Three dimensional coordinate system is divided into eight octants.
Z+
Z-X+
X-
Y+
Y-
Objective: 3.5 and 3.6 Graphing and Solving in 3D
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Plotting a Point in a 3 Dimensional System
Example: Plot (-4, 3, 4)
Z+
Z-X+
X-
Y+
Y-
Objective: 3.5 and 3.6 Graphing and Solving in 3D
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You Try
Example: Plot (3, -4, -2)
Z+
Z-X+
X-
Y+
Y-
Objective: 3.5 and 3.6 Graphing and Solving in 3D
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More Vocabulary
Linear equations in three variables is expressed as:
ax + by + cz = d
where a, b, and c are all non-zero.
The graph of a linear equation in three variables is a plane.
Example on graphing calculator.
Objective: 3.5 and 3.6 Graphing and Solving in 3D
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Graphing a Linear Equation in 3 Variables
Example: Sketch the graph of 3x + 2y + 4z = 12.
Z+
Z-X+
X-
Y+
Y-
Objective: 3.5 and 3.6 Graphing and Solving in 3D
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Steps
1. Find the x intercept by setting y and z to zero.
2. Graph the x intercept
3. Find the y intercept by setting x and z to zero.
4. Graph the y intercept
5. Find the z intercept by setting x and y to zero.
6. Graph the x intercept
7. Connect the intercepts with three lines
Objective: 3.5 and 3.6 Graphing and Solving in 3D
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You Try Graph 3x – 12y + 5z = 30
Z+
Z-X+
X-
Y+
Y-
Objective: 3.5 and 3.6 Graphing and Solving in 3D
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Solving A System in Three Variables
A system of three linear equations is a series of 3 equations with the following terms:
ax + by + cz = d
A solution to such a system is called an ordered triple.
Objective: 3.5 and 3.6 Graphing and Solving in 3D
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Types of Solutions
If the planes intersect in a single point, then the system has exactly one solution.
If the planes intersect in a line, the system has infinitely many solutions. (You’ll get an answer like 0=0)
If you get an impossible “answer” (example 0=8) then the system has no solution.
Examples!
Objective: 3.5 and 3.6 Graphing and Solving in 3D
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Solving a System Using Linear Combination
Solve: 3x + 2y + 4z = 11
2x – y + 3z = 4
5x – 3y + 5z = -1
Objective: 3.5 and 3.6 Graphing and Solving in 3D
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Steps
1. Use two of the equations and the linear combination method to rewrite the linear system in three variables as a linear system in two variables.
2. Use the third equation and one of the original equations to try to eliminate the same variable.
3. Use the linear combination method to solve the two new equations for one of the variables.
4. Plug that value into one of the “two variable” equations to get the value of the second variable.
5. Plug those two values into one of the original equations to get the third value.
Objective: 3.5 and 3.6 Graphing and Solving in 3D
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You Try Solve: x + 3y – z = -11
2x + y + z = 15x – 2y + 3z = 21
Objective: 3.5 and 3.6 Graphing and Solving in 3D
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Homework Page 173: 18-24 even, 26, 30 Page 181: 13, 15, 19, 24, 26, 36
