Yes, you can do this easily in two ways
Moving nodes along handle lines
Ensure that the nodes on the line do not have bezier style handles.
Edit the line, and select the node at the end you want to extend.
Press Ctrl+Alt and drag the node.
The node will move exactly along the line, changing the length.
This works because the handles of the nodes effectively point exactly towards the other end of the line, so dragging a node in the direction of its handle means it moves the node nearer or further away from the other end, exactly along the line itself.
Scaling the line
This works only if you have selected
Geometric bounding box as Bounding box to use in the Inkscape preferences. The Inkscape default
Visual bounding box will change the direction of your line!
turn off the scale stroke width option at the top
(as shown in the snippet below, just to the right of the 'Affect:' label)
select the line
Press the Ctrl key and drag the selection box corner at the end you
want to extend (eg where the green arrow is shown), so as to scale the line.
The line changes thichness as you drag, but snaps back when you release the mouse.
Because you do a Ctrl+Scale, you get a uniform scale without distortion.
Because the line width is not scaled, the result is a line that is longer than it was, but has the same line width - i.e. the exact same thing as extending the line.
The snapshot below shows an original line and the effect of scaling the line in either direction.