CK-12-Pre-Calculus Concepts

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

3. You could notice that the third equation is simply the sum of the other two. What happens when you do not
   notice and try to solve the system as if it were independent?
   Step 1: Rewrite the system.

x+y+z= 9

x+ 2 y+ 3 z= 22

2 x+ 3 y+ 4 z= 31

Step 2: Subtract 2 times row 1 from row 3.

x+y+z= 9

x+ 2 y+ 3 z= 22

0 x+ 1 y+ 2 z= 13

Step 3: Subtract row 1 from row 2.

x+y+z= 9

0 x+ 1 y+ 2 z= 13

0 x+ 1 y+ 2 z= 13

At this point when you subtract row 2 from row 3, all the coefficients in row 3 disappear. This means that you will
end up with the following system of only two equations and three unknowns. Since the unknowns outnumber the
equations, the system does not have a solution of one point.

x+y+z= 9

0 x+ 1 y+ 2 z= 13

Practice

1. An equation with three variables represents a plane in space. Describe all the ways that three planes could
   interact in space.


2. What does it mean for equations to be linearly dependent?


3. How can you tell that a system is linearly dependent?


4. If you have linearly independent equations with four unknowns, how many of these equations would you need in
   order to get one solution?


5. Solve the following system of equations:

3 x− 4 y+z=− 17

6 x+y− 3 z= 4

−x−y+ 5 z= 16