how this equation is a linear equation?

Here is the real exercise found on Amazon...

$$(x+5y-1)=\sqrt{2z}$$ so $$(x+5y-1)^2=2z$$ and this is not a linear equation because the order of variabes are 2.

Answer to title question: It's NOT!

Your question is legitimate:

$$x_1+5x_2-\sqrt{2x_3\;} = 1\tag{1}$$

$(1)$ is not a linear equation as you suggest.

Nor is $(f)$ linear, as typeset in the image.

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I suspect there was a misprint in the problem set (book), or a careless typo that the author (and/or editor) over-looked, and which was intended to be:

$$x_1 + 5x_2 - \sqrt{2}\;\cdot x_3 = 1\tag{2}$$

NOW, $(2)$ is a linear equation.