Answer
(a) The point $(1,-3,4)$ does not lie on the line.
(b) The point $(5,5,2)$ lies on the line.
Work Step by Step
$(x,y,z) = (2,-1,5)+n(1,2,-1)$
(a) If $(1,-3,4)$ lies on the line, then there is a real number $n$ such that $(2,-1,5)+n(1,2,-1) = (1,-3,4)$
$x$: If $2+n(1) = 1$, then $n = -1$
$y$: If $-1+n(2) = -3$, then $n = -1$
$z$: If $5+n(-1) = 4$, then $n = 1$
Since the required value of $n$ is not the same for $x,y,$ and $z$, the point $(1,-3,4)$ does not lie on the line.
(b) If $(5,5,2)$ lies on the line, then there is a real number $n$ such that $(2,-1,5)+n(1,2,-1) = (5,5,2)$
$x$: If $2+n(1) = 5$, then $n = 3$
$y$: If $-1+n(2) = 5$, then $n = 3$
$z$: If $5+n(-1) = 2$, then $n = 3$
Since the required value of $n$ is the same for $x,y,$ and $z$, the point $(5,5,2)$ lies on the line.
