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Choice A is correct. The second equation in the given system defines the value of $x$ as $5y$. Substituting $5y$ for $x$ into the first equation yields $5y+y=18$ or $6y=18$. Dividing each side of this equation by $6$ yields $y=3$. Substituting $3$ for $y$ in the second equation yields $5\left(3\right)=x$ or $x=15$. Therefore, the solution $\left(x,y\right)$ to the given system of equations is $\left(15,3\right)$. 

Choice B is incorrect. Substituting $16$ for $x$ and $2$ for $y$ in the second equation yields $5\left(2\right)=16$, which is not true. Therefore, $\left(16,2\right)$ is not a solution to the given system of equations.

Choice C is incorrect. Substituting $17$ for $x$ and $1$ for $y$ in the second equation yields $5\left(1\right)=17$, which is not true. Therefore, $\left(17,1\right)$ is not a solution to the given system of equations.

Choice D is incorrect. Substituting $18$ for $x$ and $0$ for $y$ in the second equation yields $5\left(0\right)=18$, which is not true. Therefore, $\left(18,0\right)$ is not a solution to the given system of equations.