3.5- G RAPHING L INEAR E QUATION IN 3 V ARIABLES.

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Presentation transcript:

3.5- G RAPHING L INEAR E QUATION IN 3 V ARIABLES

T HE 3D C OORDINATE S YSTEM

O RDERED T RIPLE 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)

P LOTTING O RDERED T RIPLE Plot the following points: 1. (3, -1, -5) 2. (-2, 3, 4)

X-, Y-, AND Z- I NTERCEPTS 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)

F IND ALL 3 INTERCEPTS. 1. 3x – 12y + 5z = x – 2y + 2z = 6

G RAPHING 3D F UNCTIONS Standard Form: ax + by + cz = d, where a, b, c, and d are any real number. Steps to Graph: 1. Find all 3 intercepts 2. Plot the 3 intercepts 3. Connect and shade the result

G RAPH THE F OLLOWING F UNCTIONS -x + 3y + 4z = 12

G RAPH THE F OLLOWING F UNCTIONS 5x = y – 5z

– 2z = -18 – 6x – 3y

R EWRITE 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.

S TEPS 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) = ”

R EWRITE SO Z IS A FUNCTION OF X AND Y. T HEN EVALUATE F (-2, 2). 1.3x – 12y + 5z = 30

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

S YSTEMS 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!

E XAMPLES – FIND THE SOLUTION TO THE SYSTEM OF EQUATIONS. 1.x + 3y – z = -11 2x + y + z = 1 5x – 2y + 3z = 21 A. (1, -4, 0) B. (1, 0, -1) C. (2, -4, 1) D. (2, 4, -1)

E XAMPLES – FIND THE SOLUTION TO THE SYSTEM OF EQUATIONS. 2.x – 3y + 6z = 21 3x + 2y – 5z = -30 2x – 5y + 2z = -6 A. (3, -8, -1) B. (-3, 2, 5) C. (3, 8, -1) D. (-3, -2, 5)

HOMEWORK Page 174 # 38 – 41 Page 181 # 5 – 7