10.5#10

On problem 10 of 10.5, two parametric equations are given. To solve for the intersection we solve for two variables, usually t and s in the equations. Unfortunately these equations have only t’s. The problem can be solved by assuming the equations for L2 use a different variable, independent from t.

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No typo, really, but you make a useful point.  It's easy to make the mistake of thinking that the t in the parametric equation of a line refers to anything real.  It doesn't, it's just a dummy variable.  It's just a way of labeling points on the line. For example L1: x=2+12t,   y=6+8t,  z=4+2t is another labeling that would do just as well, and the dummy variable associated t associated with this labeling would be different.

It's true that if you expect t to be a real thing and look for an intersection by setting 14+6t=-18+7t, 14+4t=-10+6t, 6+t=-5+4t, there will almost always be no such t.  And yes, the correct way to look for an intersection would be to allow give two different names to the parameters for the two different lines. The names t and s would work as well as any other two names.

(to misquote Wild Bill Shakespeare, a variable by any other name would be just as good, or smell as sweet.  Wild Bill had much to say on this subject, btw:

JULIET

'Tis but thy name that is my enemy;
Thou art thyself, though not a Montague.
What's Montague? it is nor hand, nor foot,
Nor arm, nor face, nor any other part
Belonging to a man. O, be some other name!
What's in a name? that which we call a rose
By any other name would smell as sweet;
So Romeo would, were he not Romeo call'd,
Retain that dear perfection which he owes
Without that title. Romeo, doff thy name,
And for that name which is no part of thee
Take all myself.

)