Given an equation $a^2=b^2$, solving for $a$ would give
$$\sqrt {a^2} = \sqrt {b^2}$$
$$|a| = |b|$$
$$a = \pm |b|$$

Is there a way for me to solve for $a$ without having the absolute value sign on $b$? I believe the answer should be $a = \pm b$ and this is my goal of doing all this. If I divide $a = \pm |b|$ into cases and try to solve for b:

Case one: $a = |b|$
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$b = a$ or $b = -a$. Put it another way, $b = \pm a$

Case two: $a = -|b|$
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Solving for $|b|$ would give us $|b| = -a$. Hence,

$b = (-a)$ or $b = -(-a)$. Put this another way, $b = \pm (-a)$

So, now I have $b = \pm a$ or $b = \pm (-a)$. I believe if I try to solve for $a$ on both equations, I'll get (correct me if I'm wrong):
$$a = \pm b$$
or
$$a = \pm (-b)$$
So, is it correct if I say that solving for $a$ in $a^2 = b^2$ gives me $a = \pm b$? Even though it actually gives me $a = \pm b$ or $a = \pm (-b)$. If yes, how so since there are two possibilities. I'm also aware that $\pm b \neq \pm (-b)$ which makes me even more confuse about it. I appreciate any help.