<s>[INST] Give the equation of a line that passes through the point (5,1) that is perpendicular and parallel to line A. The equation of line $A$ is $3x + 6y - 1 = 0$. Give the equation of a line that passes through the point $(5,1)$ that is


*

*Perpendicular to line $A$.

*Parallel to line $A$.
Attempting to find the parallel,
I tried $$y = -\frac{1}{2}x + \frac{1}{6}$$
$$y - (1) = -\frac{1}{2}(x-5)$$
$$Y = -\frac{1}{2}x - \frac{1}{10} - \frac{1}{10}$$
$$y = -\frac{1}{2}x$$
 [/INST] Line perpendicular to A that passes through the point $(5,1)$: it is directed by the normal vector to A: $(3,6)$ \ or $(1,2)$. Hence its equation is
$$\frac{x-5}1=\frac{y-1}2\iff 2x-y-9=0. $$
Line parallel to A:it has the smame normal vector as A:
$$3x+6y=3\cdot 5+6\cdot 1\iff 3x+6y-21=0.$$
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