How do you solve a systems of equations using rref when the system in dependent?

Question

In my case, after I used the rref function on my calculator to solve the equations:

x − y + 7z = 3
x − y + 8z = 2

 

It gave the following as a matrix:

1  -1  0  10

0  0  1  -1

 

I think this means the systems are dependent. How do I express the answer as coordinates (x, y, z)?

Answer 1

Subtract the top equation from the bottom equation and see that 

 

z = -1

 

Plug that into both equations and get

 

x - y - 7 = 3 ⇒ x - y = 10

x - y - 8 = 2 ⇒ x - y = 10

 

The 2 equations are the same.  You can express y in terms of x, giving infinitely many solutions, with x being any real number.

 

x = any real number

y = x - 10

z = -1