1.
In the field \(\Z_{5}\) (with operations \(+\) and \(\cdot\)), you might solve the equation below as shown.
\begin{align*}
x\cdot 4 + 2 \amp = 1 + x\\
3\cdot x + 2\amp = 1 \\
3\cdot x \amp = 4\\
x \amp = 3
\end{align*}
This actually uses more steps than are written above. In fact, this only works because all nine field axioms hold for \(\Z_{5}\text{.}\)
Demonstrate where each field axiom is used and how it is applied by solving the equation using more steps, citing each axiom as you go. You may assume that \(0x = 0\) without proof.
Hint.
A good way to make this clear is to give a sequence of equations where each follows from the previous using exactly one field axiom. For example, you might start like this:
\begin{align*}
x\cdot 4 + 2 \amp = 1 + x\\
(x\cdot 4 + 2) + 4\cdot x\amp = (1 + x) + 4\cdot x \amp \text{add } 4\cdot x \text{ to both sides}\\
(x\cdot 4 + 2) + 4\cdot x\amp = 1 + (x + 4\cdot x) \amp \text{addition is associative}
\end{align*}
I would expect at least 15 more lines after these to complete the solution.
