These are simply the vector equation separated by components:$$(x,y)=(p_1,p_2)+k\cdot (v_1,v_2)$$ $$$ \left. \begin{array}{rcl} x &=&p_1+k\cdot v_1 \\ y &=& p_2+k\cdot v_2 \end{array}\right \}$$$
Find the parametrical equations of the straight line $$r$$ that crosses the points $$(3, 4)$$ and $$(-2, 6)$$.
The vector equation with $$A=(3,4)$$ and $$B=(-2,6)$$ is: $$$(x, y) = A + k \cdot \overrightarrow {AB} = (3, 4) + k \cdot (-5, 2)$$$ Therefore, the parametrical equations of the straight line are: $$$\left. \begin{array}{rcl} x=3-5 \cdot k \\ y=4+2 \cdot k \end{array} \right\}$$$