How where to practice solving systems of equations over finite fields

Question

so in college I am taking Linear Algebra and we're having a test. In this test we have to solve a system of equations over a finite field.

Let's have a finite field $Z_n = \{0, 1, ..., n - 1\}$. So for example a field $Z_3 = \{0, 1, 2\}$.

Now let's say that we have to solve the system of equations below in $Z_3$.

$x + 2y + z = 1 \\ x + 2y + 2z = 1 \\ 2x + y + z = 1$

As I am farily new to this I am not confident enough solving this. However I could not find any website which would provide practice problems (with solutions) like these.

Could you please suggest me where I'd find some?

Thank you

Answer 1

$$x + 2y + z = 1 \\ x + 2y + 2z = 1 \\ 2x + y + z = 1$$

Add the first two equations and we get $$ 2x+y=2$$ Plug in the third equation we get $z=2 $

Subtracting the first equation from the second results in $z=0$

Therefore the system as given is inconsistent.

However if we change the middle equation to $$x+y+2z=1$$ then we have $$x=y=z=1$$

Answer 2

A commonly taught method for solving systems of linear equations over a field is Gaussian elimination. It is a simple algorithm that works over any field, also finite fields.