Absolute Value equations in 2 variables ($|x-y|+|x+y|=\sqrt5$) [duplicate]

Solution 1:

$|x-y|+|x+y|=\sqrt5$

We have four possible cases -

$(i) ~ ~ x-y \geq 0, x + y \geq 0$

which is equivalent to $ ~ -x \leq y \leq x, x \geq 0$

That leads to $x = \frac{\sqrt5}{2}$

Similarly,

$(ii) ~ ~ x-y \geq 0, x + y \leq 0$

leads to $ ~ y = - \frac{\sqrt5}{2}$

$(iii) ~ ~ x-y \leq 0, x + y \geq 0$

leads to $y = \frac{\sqrt5}{2}$

$(iv) ~ ~ x-y \leq 0, x + y \leq 0$

leads to $ ~ x = - \frac{\sqrt5}{2}$