Simple modulus question

Question

Probably a very simple question but not 100% sure.

If $$\lvert x-y \rvert = \lvert y-z \rvert$$

am I right in saying that

$$z=x$$

?

Thanks

Answer 1

Take a positive value, say $7$. The equation $|x-y|=7$ represents two right lines, $L_1$ and $L_2$, of common slope $1$ passing by $(0,7)\in L_1$ and $(0,-7)\in L_2$ and this is so (changing notation in coordinates) for $|y-z|=7$. Take now, for example, $y=2$ for these two lines so you have $$|x-2|=|2-z|=7$$ with $x=-5\ne z=9$

Answer 2

Two numbers with the same absolute value are equal or of opposite signs.

$$|a|=|b|\iff a=\pm b$$ hence

$$x-y=\pm(y-z),$$

$$x-2y+z=0\lor x=z.$$

Answer 3

No we can’t make such an conclusion. We can know that x-y = y-z or x-y=z-y. When x-y =z-y. We can say x=z. But when x-y=y-z, y=(x+z)/2. We can’t say x = z.