prove $-a = (-1)a$ ? I've done two possible proofs. Which is valid if at all?

Question

I'm self studying math. I'm just starting "A Survey of Modern Algebra" by Birkhoff and Mac Lane because I thought the proof section of this book was more clear than Shilov's first chapter in "Introduction to Real and Complex Analysis". I'm pretty new to proofs. I know this is perhaps very simple but I need some feedback since there are no answers in the back and I have no one else to talk to about proofs. Are either of these acceptable? Both? Thank you for your patience.

page 8 Problem 3 (d) : Prove $-a = (-1)a$

proof one

let $a = 1 $

$-1 = -1(1) $ by substitution

$-1 = -1 $ by unity

proof two

since $a = 1a$

$(-1)a = (-1)a$ by substitution on the left

Answer 1

None of your proofs are actually proofs.

Note that $ -a$ is the additive opposite of $ a $ Therefore to show that $(-1)a=-a$, we have to prove that $$(-1)a+a=0$$

Since $a=(1)a$, we have $$ (-1)a + a = (-1)a + (1)a = (-1+1)a = 0a =0$$

Thus $(-1)a$ is indeed the additive inverse of $a$, that is $(-1)a=-a$

Answer 2

They are both wrong. The first one is nonsense. In second one, you don't even deduce what you were supposed to deduce.

Note that\begin{align}(-1)a+a&=(-1)a+1a\\&=\bigl((-1)+1\bigr)a\\&=0\times a=0.\end{align} Therefore, $(-1)a=-a$, since each element has one and only one inverse for the addition.