Sqrt of y^2=x^2: Why y=x or y=-x?

[Mr Davis 97]

Say we have that $y^2 = x^2$. Then if we take the square root of both sides, it would seem that we have $|y| = |x|$. Why does this imply that that $y=x$ or $y=-x$, rather than implying that $y=|x|$ or $y=- |x|$?

 

[Andrew Mason]

Say we have that $y^2 = x^2$. Then if we take the square root of both sides, it would seem that we have $|y| = |x|$. Why does this imply that that $y=x$ or $y=-x$, rather than implying that $y=|x|$ or $y=- |x|$?

The two answers are equivalent.

We agree that a = |x| means that
a = x or
a = -x.

If you let a = |y| then
(1) |y| = x or
(2) |y| = -x

But (1) x = |y| means x = y or x = -y and (2) -x = |y| means -x = y or -x = -y (i.e. x = y). So, y = x or y = -x.

AM

 

[Mark44]

Say we have that $y^2 = x^2$. Then if we take the square root of both sides, it would seem that we have $|y| = |x|$. Why does this imply that that $y=x$ or $y=-x$, rather than implying that $y=|x|$ or $y=- |x|$?

Why not just do this?
$y^2 = x^2 \Leftrightarrow y^2 - x^2 = 0 \Leftrightarrow (y - x)(y + x) = 0 \Leftrightarrow y = x \text{ or } y = -x$

 

[mfb]

Yet another approach:
If |x|=|y|, there are four cases:
x and y positive: then x=y
x positive, y negative: x=-y
x negative, y positive: x=-y
x and y negative: x=y
Combined, x=y or x=-y. In other words, the two variables are identical up to a possible difference in their sign.

I neglected the option x=y=0 here, but that fits to the answer as well.