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}$