Why does a first order reaction only depend on the concentration of a single reactant?

I am new to kinetics so please explain it from basic, If there is a reaction -

$$\ce{A + B + C -> D}$$

In first order reaction rate is given by-

$$\mathrm R = k[\mathrm A]$$ or $$\mathrm R = k[\mathrm B]$$ or $$\mathrm R = k[\mathrm C]$$

my question is why cant $$\mathrm R = k[\mathrm A]^{1/3}[\mathrm B]^{1/3}[\mathrm C]^{1/3}$$ even though order of reaction is still $$1$$,----- is it by definition that 1st order reaction always depend on concentration of single reactant, or is there any experimental proof for this?.

• because the definition of first order reaction is that it depends on concentration of single reactant – chemstackisunhelpful Jun 2 at 12:55
• $\ce{A + B + C -> D}$ is not a first order reaction. – Martin - マーチン Jun 2 at 13:16
• No. If $\ce{A + B + C -> D}$, then the rate law would be $r = k[\ce{A}][\ce{B}][\ce{C}]$, if the rate law is determined to be $r = k[\ce{A}]$, then the reaction cannot be elementary. – Martin - マーチン Jun 2 at 17:15