Question 202834
The equation of the line that contains the point (-4, -10) and is perpendicular to the line that contains the points (-3, 6)and (3, 12) is? 
a. Y= x+9 
b. Y= -x-14 
c. Y= -x+9 
d. Y= 16x+54

Ok, let's start this from the idea that the equation of the line must be perpendicular to the points (-3, 6)and (3, 12).   There are two facts you must know to solve this problem:


1)  Perpendicular lines have slopes that are NEGATIVE reciprocals of one another. That's just a fancy way of saying you "flip" them and take the opposite.   Sooo, if your slope is {{{3/4}}}, a line perpendicular to that is {{{-3/4}}}. 



2)  To find the slope of a line, you follow this formula:


{{{y[2]-y[1]}}}
______________
{{{x[2]-x[1]}}}

(I am very sorry this fraction does not look nice.  I am not adept yet at making fractions and what-not on this site).  



So, let's find the slope of these ordered pairs:


(-3, 6)and (3, 12)


{{{12-6}}}
__________
{{{3- -3}}}, which is:  


{{{12-6}}}
__________
{{{3+3}}} which is:


{{{6/6}}} = 1 or {{{1/1}}}



What is the negative reciprocal of {{{1/1}}}?  
The negative reciprocal is:  {{{-1/1}}} or:  -1



Now we must make sure that the equation of the line includes the point (-4, -10) and must have a slope of -1 if it is to be perpendicular to the line that contains the points (-3, 6)and (3, 12).


To find this equation of a line, you would need the POINT SLOPE equation of a line.  In other words, you know a point on the line (-4, -10) and you know the slope is -1 sooo, you just fill that info into the point slope equation of a line.  


Let's look at the POINT SLOPE equation of a line.   In this equation, "m" is the slope and {{{y[1]}}} and {{{x[1]}}} refer to a point that the line goes thru.  Here is the point slope equation of a line:


{{{y-y[1] = m(x - x[1])}}}


Now just fill in with this info:
m = -1
{{{x[1] = -4}}}
{{{y[1] = -10}}}


{{{y-y[1] = m(x - x[1])}}} (point slope equation of line)
{{{y -  -10 = -1(x - -4)}}} (fill in given values, which computes to...)
{{{y +10 = -1(x+4)}}} (<---which computes to this line)
{{{y +10 = -1x - 4}}} (distribute the -1 to the "x" and to the 4)
{{{y = -1x - 4 -10}}}  (subtract -10 from both sides)
{{{y = -1x -14}}} (final answer)


Therefore, your answer is "b" which is: b. Y= -x-14 

I hope this helps you! :-)