1. Section 3.5 - Graphing Linear Equations in Three Variables
ALGEBRA TWO
CHAPTER THREE: SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES
Section Graphing Linear Equations in Three Variables

2. LEARNING GOALS
Goal One - Graph linear equations in three variables and evaluate linear functions of two variables.
Goal Two - Use functions of two variables to model real-life situations.

3. VOCABULARY
A linear equation in three variables, x, y, and z is an equation of the form
ax + by + cz = d, where a, b, and c are all not zero.
In a three-dimensional coordinate system, the xy-plane is in a horizontal position, with the z-axis as a vertical line through the origin. 1

4. VOCABULARY
An ordered triple is given by (x, y, z) and represents a point in space. 1

6. Graphing a 3 variable equation
STEP 1a: Find the points where the graph intersects the axes. Let y = 0 and z = 0, and solve for x.
STEP 1b: Find the points where the graph intersects the axes. Let x = 0 and z = 0, and solve for y.
STEP 1c: Find the points where the graph intersects the axes. Let x = 0 and y = 0, and solve for z.
STEP 2: Plot the intercepts on an xyz- coordinate plane.
STEP 3: Draw lines to connect the points.
STEP 4: Shade in the triangle formed by the lines.

7. Sketch the graph of -2x + 4y + 8z = 16
Graphing a Linear Equation in Three Variables
Sketch the graph of -2x + 4y + 8z = 16
STEP 1a: Find the points where the graph intersects the axes. Let y = 0 and z = 0, and solve for x.
-2x + 4y + 8z = 16
The x-intercept is the point (-8, 0, 0)
-2x + 4(0) + 8(0) = 16
-2x/-2 = 16/-2
x = -8 1

8. Sketch the graph of -2x + 4y + 8z = 16
Graphing a Linear Equation in Three Variables
Sketch the graph of -2x + 4y + 8z = 16
STEP 1b: Find the points where the graph intersects the axes. Let x = 0 and z = 0, and solve for y.
-2x + 4y + 8z = 16
The y-intercept is the point (0, 4, 0)
-2(0) + 4y + 8(0) = 16
4y/4 = 16/4
y = 4 1

9. Sketch the graph of -2x + 4y + 8z = 16
Graphing a Linear Equation in Three Variables
Sketch the graph of -2x + 4y + 8z = 16
STEP 1c: Find the points where the graph intersects the axes. Let x = 0 and y = 0, and solve for z.
-2x + 4y + 8z = 16
The z-intercept is the point (0, 0, 2)
-2(0) + 4(0) + 8z = 16
8z/8 = 16/8
z = 2 1

10. Sketch the graph of -2x + 4y + 8z = 16
Graphing a Linear Equation in Three Variables
Sketch the graph of -2x + 4y + 8z = 16
STEP 2: Plot the intercepts on an xyz- coordinate plane.
The x-intercept is the point (-8, 0, 0)
The y-intercept is the point (0, 4, 0)
The z-intercept is the point (0, 0, 2) 1

11. Sketch the graph of -2x + 4y + 8z = 16
Graphing a Linear Equation in Three Variables
Sketch the graph of -2x + 4y + 8z = 16
STEP 3: Draw lines to connect the points. 1

12. Sketch the graph of -2x + 4y + 8z = 16
Graphing a Linear Equation in Three Variables
Sketch the graph of -2x + 4y + 8z = 16
STEP 4: Shade in the triangle formed by the lines. 1

13. Evaluate a function in two variables
Step 1: Solve the equation for z.
Step 2: Substitute the given values of x and y
Step 3: Simplify to find z ( or f(x,y) )

14. Evaluating a Function of Two Variables (1) Solve the equation for z.
Write the linear function 10x - 3y + 12z = - 60 as a function of x and y.
Evaluate the function for f (3, -2)
(1) Solve the equation for z.
10x - 3y + 12z = -60
Write original equation
12z = -10x + 3y + -60
Bring 10x and 3y to right side.
z = (-5/6)x + (1/4)y - 5
Divide everything by 12
f (x, y) = (-5/6)x + (1/4)y - 5
Replace z with f (x, y) 1

15. Evaluating a Function of Two Variables
Write the linear function 10x - 3y + 12z = - 60 as a function of x and y.
Evaluate the function for f (3, -2)
(2) Substitute 3 for x and -2 for y.
f (3, -2) = (-5/6)(3) + (1/4)(-2) - 5
Substitute
f (3, -2) = (-5/2) + (-1/2) - 5
Multiply
f (3, -2) = (-6/2) - 5
Add fractions
The graph contains the point (3, -2, -8)
Simplify
f (3, -2) = (-3) - 5
f (3, -2) = -8
Simplify 1

16. ASSIGNMENT
READ & STUDY: pg
WRITE: pg #27, #31, #37, #39, #45, #51