This is not a question on my homework, just one from the book I'm trying to figure out. They want me to find the intersection of these two lines: \begin{align} L_1:x=4t+2,y=3,z=-t+1,\\ L_2:x=2s+2,y=2s+3,z=s+1. \end{align} But they do not provide any examples. Flipping to the back it tells me that they do intersect and at the point $(2,3,1).$ How did they arrive at this answer? I'm just hoping to understand because I cannot derive any answer.
Thank you for your time,
They intersect each other when all their coordinates are the same.
If we call $L_1=\langle x_1,y_1,z_1\rangle$ and $L_2=\langle x_2,y_2,z_2\rangle$ then you have to solve the system: $$x_1=x_2\Longrightarrow4t+2=2s+2,$$ $$y_1=y_2\Longrightarrow3=2s+3,$$ $$z_1=z_2\Longrightarrow1-t=s+1.$$
In this case, if we set both parameters equal to zero, the system will be solved. $$x_1=x_2\Longrightarrow2=2,$$ $$y_1=y_2\Longrightarrow3=3,$$ $$z_1=z_2\Longrightarrow1=1.$$
The system is solved for $t=0=s$. When you plug $t=0$ in $L_1$ you get $\langle 2,3,1\rangle$. The same happens when you plug $s=0$ in $L_2$.