3.5- GRAPHING LINEAR EQUATION IN

3 VARIABLES

THE 3D COORDINATE SYSTEM

ORDERED TRIPLE

Before, with 2 dimensions, there was just an x and a y coordinate and you had ordered pairs (x, y)

Now with 3 dimensions, there are ordered triples: (x, y, z)

PLOTTING ORDERED TRIPLE

Plot the following points:1. (3, -1, -5)2. (-2, 3, 4)

X- , Y- , AND Z- INTERCEPTS

X-Intercept: the point where the graph crosses the x – axis. In the form of (x, 0, 0)

Y-Intercept: the point where the graph crosses the y – axis. In the form of (0, y, 0)

Z-Intercept: the point where the graph crosses the z – axis. In the form of (0, 0, z)

FIND ALL 3 INTERCEPTS.

1. 3x – 12y + 5z = 30

2. x – 2y + 2z = 6

GRAPHING 3D FUNCTIONS

Standard Form: ax + by + cz = d, where a, b, c, and d are any real number.

Steps to Graph:1. Find all 3 intercepts2. Plot the 3 intercepts3. Connect and shade the result

GRAPH THE FOLLOWING FUNCTIONS

-x + 3y + 4z = 12

GRAPH THE FOLLOWING FUNCTIONS

5x = 15 + 3y – 5z

– 2z = -18 – 6x – 3y

REWRITE THE EQUATION AS A FUNCTION OF X AND Y.

This means to get z by itself.

It is denoted as “f(x, y)” rather than “z = ”

The reason is so you can substitute numbers for x and y and evaluate z.

STEPS TO REWRITE Z AS A FUNCTIONS OF X AND Y

1. Add or Subtract the ‘x’ and ‘y’ terms to the other side of the equal sign.

2. Divide every term by the constant “attached” to z

3. Now since “z = ” is all you have on the left side replace z with “f(x, y) = ”

REWRITE SO Z IS A FUNCTION OF X AND Y.THEN EVALUATE F(-2, 2). 1. 3x – 12y + 5z = 30

REWRITE SO Z IS A FUNCTION OF X AND Y.THEN EVALUATE F(0, -2).2. x – 2y + 2z = 6

SYSTEMS WITH 3 EQUATIONS

If you have a system with 3 equations, there are 3 variables you need to solve for.

These questions will be multiple choice, but the answer you pick must satisfy ALL 3 EQUATIONS!

EXAMPLES – FIND THE SOLUTION TO THE SYSTEM OF EQUATIONS.

1. x + 3y – z = -112x + y + z = 15x – 2y + 3z = 21

A. (1, -4, 0)B. (1, 0, -1)C. (2, -4, 1)D. (2, 4, -1)

EXAMPLES – FIND THE SOLUTION TO THE SYSTEM OF EQUATIONS.

2. x – 3y + 6z = 213x + 2y – 5z = -302x – 5y + 2z = -6

A. (3, -8, -1)B. (-3, 2, 5)C. (3, 8, -1)D. (-3, -2, 5)

HOMEWORK

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