Find the value of $|a|+|b|$

Question

Find the value of $|a|+|b|$ such that the identity >$$|ax+by|+|bx+ay|=|x|+|y|$$ hold for all $x,y\in\mathbb{R}.$

Try:put $x=0,$ Then $|y|\bigg(|a|+|b|\bigg)=|y|$ means either $|y|=0$ or $|a|+|b|=1$

:put $y=0,$ Then $|x|\bigg(|a|+|b|\bigg)=|x|$ means either $|x|=0$ or $|a|+|b|=1$

So $|a|+|b|=1$

Could some help me is there is any value of $|a|+|b|$ exists.thanks

Answer 1

If the identity holds for all $x,y$, it certainly holds for $x=1,y=0$ and it reduces to $\color{green}{|a|+|b|=1}$, which is your answer.

Then if we pick $a=1,b=0$, we do have

$$|ax+by|+|bx+ay|=|x|+|y|$$

true for all $x,y$.

As nothing specific was asked about the individual values of $a,b$, we don't need to care about the existence of other solutions.

Answer 2

Hint: $\;x=y=1 \implies 2|a+b| = 2\,$, and $\,x=1, y=-1 \implies 2|a-b| = 2\,$. But $\,|a+b|=|a-b|\,$ can only hold if either $\,a=0\,$ or $\,b=0\,$, and then the other one must be $\,\pm1\,$.