Is x=2 the Correct Solution to x+5=1 in ℤ6?

[Math9999]

Homework Statement

Solve x+5=1 in ℤ₆.

Homework Equations

None.

The Attempt at a Solution

6+1=7
7-5=2
Is this the right way to solve this problem?

 

[fresh_42]

I'm not sure what your thoughts were here, so I don't know, whether this is the right way. There are more than one way to get the correct result. In $\mathbb{Z}_6$ there is neither an element $6$ nor $7$; at least not in its usual representation. But you can use the normal way to solve such equations:
$$
x+5=1\,\vert \,-5 \\
x = 1 - 5 \,=\, -4
$$
and then determine what $-4 \in \mathbb{Z}_6$ is.

 

[Math9999]

It's modular arithmetic, the answer is x=2.