Read a common Cartesian parameter and rescale it consistently

Let $(x-5)/3=1-y=2z+4=\lambda$, with $\lambda\in\mathbb R$. Then $(x-5)/3=\lambda$ gives x=5+3λ; 1−y=λ gives y=1−λ; and 2 z+4=λ gives z=−2+λ/2.

Therefore $\mathbf r=(5,1,-2)+\lambda(3,-1,1/2)$, $\lambda\in\mathbb R$.

The same line is $\mathbf r=(5,1,-2)+\lambda'(6,-2,1)$, with $\lambda'=\lambda/2$. Both parameters range over all real values. Converting between Cartesian, parametric and vector forms preserves the line, even when the direction vector and parameter change together.