Consider the points $$A = (2, 1,-2)$$ and $$B = (1,-2, 3)$$, and find the implicit equations of the straight line that goes through $$A$$ anb $$B$$.
Development:
We will start computing a director vector of the straight line: $$$\overrightarrow{AB}=B-A=(1,-2,3)-(2,1,-2)=(-1,-3,5)$$$
Therefore, with the director vector and point $$A$$, we obtain the continuous equation: $$$\dfrac{x-2}{-1}=\dfrac{y-1}{-3}=\dfrac{z+2}{5}$$$
Finally, if we separate the continuous equations and simplify a little bit we have: $$$\dfrac{x-2}{-1}=\dfrac{y-1}{-3} \Rightarrow -3x+6=-y+1 \Rightarrow -3x+y+5=0$$$ $$$\dfrac{x-2}{-1}=\dfrac{z+2}{5} \Rightarrow 5x-10=-z-2 \Rightarrow 5x+z-8=0$$$ Therefore the implicit equations are: $$$-3x+y+5=0$$$ $$$5x+z-8=0$$$
Solution:
$$-3x+y+5=0$$; $$5x+z-8=0$$