CK-12-Pre-Calculus Concepts

8.2. Systems of Three Equations and Three Unknowns http://www.ck12.org

x+ 2 y−z= 0

0 x+y+z= 10

0 x+ 0 y+ 22 z= 154

Step 4: Divide row 3 by 22.

x+ 2 y−z= 0

0 x+y+z= 10

0 x+ 0 y+z= 7

Step 5: Subtract row 3 from row 2

x+ 2 y−z= 0

0 x+y+ 0 z= 3

0 x+ 0 y+z= 7

Step 6: Add row 3 to row 1

x+ 2 y+ 0 z= 7

0 x+y+ 0 z= 3

0 x+ 0 y+z= 7

Step 7: Subtract 2 times row 2 to row 1.

x+ 0 y+ 0 z= 1

0 x+y+ 0 z= 3

0 x+ 0 y+z= 7

The solution to the system is (1, 3, 7) exactly as it should be.
Example B
Is the following system linearly independent or dependent? How do you know?

3 x+ 2 y+z= 8

x+y+z= 3

5 x+ 4 y+ 3 z= 14

6 x+ 6 y+ 6 z= 18