How many solutions are there in Z2?

[loki91]

Homework Statement

In Z2 find all solutions to

x1 + x3 + x4 = 1
x1 +x2 +x4 +x5 =1
x1 + x5 = 1.

How many solutions in total are there?

The Attempt at a Solution

I attempted to reduce row it however I'd be left with a "1" on the right hand side. Its not hard but I am hitting some sort of block!

 

[vela]

The presence of a 1 on the RHS isn't a problem. You're trying to solve an inhomogeneous system, so a constant term in the solution is expected.

What equations did you get after you reduced them?

 

[Mark44]

Since this is a nonhomogeneous system (as vela points out), you should be working with an augmented matrix with 3 rows and 6 columns. The sixth column will have the constants.

 

[loki91]

The presence of a 1 on the RHS isn't a problem. You're trying to solve an inhomogeneous system, so a constant term in the solution is expected.

What equations did you get after you reduced them?

Since this is a nonhomogeneous system (as vela points out), you should be working with an augmented matrix with 3 rows and 6 columns. The sixth column will have the constants.

Okay, after I reduced this is what I got:

1 0 0 0 1 1
0 1 0 1 0 0
0 0 1 1 -1 0

so,

x3 = x5 - x4
x2 = - x4
x1 = - 1 - x5

??

 

[vela]

It should be x1 = 1 - x5, but in Z₂, it doesn't really matter.

Now just enumerate all possible combinations for x4 and x5 and evaluate the equations for each combination to find all the solutions to the system.