<s>[INST] Equations of the line which intersects the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4}$ Find the equations of the line which intersects the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4}$  and passes through the point $(1,1,1)$.
First I thought required line passes through point on intersection of given lines, but as I checked, lines are non-intersecting. So doesn't that mean that there will be infinite lines which satisfy the given conditions?
 [/INST] Can't comment:
To get to the result that J. Wu and Dr. Sonnard Graubner gets, you can assume that everything is equal to $t$ or seperately
$$t=\frac{x-1}{2}; t=\frac{y-2}{3};t=\frac{z-3}{4}$$
and solve for $x,y,$ and $z$ in each seperate equations
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