Question

Is each line parallel, perpendicular, or neither parallel nor perpendicular to a line whose slope is −6?

Answer 1

Line M with slope 6 is neither, line n with slope -6 is parallel because only lines with same slope are parallel, line p with slope 1/6 is neither. and line q with slope -1/6 is perpendicular because only the reciprocal of a slope is perpendicular, so reciprocal of 6= 1/6 reciprocal of -6=-1/6 reciprocal of 2= 1/2

Answer 2

we have

the slope of the given line is

$m1=-6$

we know that

If two lines are parallel , then their slopes are the same

so

$m1=m2$

if two lines are perpendicular, then the product of their slopes is equal to minus one

so

$m1*m2=-1$

we will proceed to verify each case to determine the solution

case A) line m with slope $6$

Compare the slope of the line m of the case A) with the slope of the given line

$m1=-6$  -----> slope given line

$m2=6$ ----> slope line m case A)

$m1\neq m2$

$m1*m2\neq-1$

therefore

the line m case A) and the given line are neither parallel nor perpendicular

case B) line n with slope $-6$

Compare the slope of the line n of the case B) with the slope of the given line

$m1=-6$  -----> slope given line

$m2=-6$ ----> slope line n case B)

$m1=m2$ ------> the lines are parallel

case C) line p with slope $\frac{1}{6}$

Compare the slope of the line p of the case C) with the slope of the given line

$m1=-6$  -----> slope given line

$m2=\frac{1}{6}$ ----> slope line p case C)

$m1*m2=-6*\frac{1}{6}=-1$ ------> the lines are perpendicular

case D) line q with slope $-\frac{1}{6}$

Compare the slope of the line q of the case D) with the slope of the given line

$m1=-6$  -----> slope given line

$m2=-\frac{1}{6}$ ----> slope line q case D)

$m1\neq m2$    

$m1*m2\neq-1$

therefore

the line q case D) and the given line are neither parallel nor perpendicular

the answer in the attached figure