SOLUTION: A line passing through (–6, –5) and (–1, y) is perpendicular to a line with slope -5/13 . Find the value of y.

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Question 111717: A line passing through (–6, –5) and (–1, y) is perpendicular to a line with slope -5/13 . Find the value of y.
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
A line passing through (–6, –5) and (–1, y) is perpendicular to a line with slope -5/13 . Find the value of y.

to find the value of y, you first need to use the given data that this line is perpendicular to a line with slope m%5B1%5D+=-5%2F13, so you can calculate a slope m%5B2%5D for your line
we know that if the lines are perpendicular, then : m%5B1%5D+m%5B2%5D=+-+1
find slope for line:

m%5B1%5D+m%5B2%5D=+-+1

m%5B2%5D+=+-+1%2Fm%5B1%5D

m%5B2%5D+=+-+1%2F%28-5%2F13%29

m%5B2%5D+=+-+13%2F-5

m%5B2%5D+=+13%2F5 ….this is a slope of a line passing through (–6, –5) and (–1, y)

we can calculate y using this formula:
m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28+x%5B2%5D+-+x%5B1%5D%29

13%2F5+=+%28y+-+%28-5%29%29%2F%28-1+-+%28-6%29%29

13%2F5+=+%28y+%2B+5%29%2F%28-1+%2B+6%29
13%2F5=+y%2F5+%2B+1…………………………move 1 to the left
13%2F5+%96+1+=+y%2F5………multiply both sides by 5
13%2A5%2F5+%96+1%2A5+=+y%2A5%2F5………
13+%96+5+=+y………
8+=+y………,,,,,,,,,,,,,,,,,,
Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (-6,-5) and (-1,8)


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: (x%5B1%5D,y%5B1%5D) is the first point (-6,-5) and (x%5B2%5D,y%5B2%5D) is the second point (-1,8))


m=%288--5%29%2F%28-1--6%29 Plug in y%5B2%5D=8,y%5B1%5D=-5,x%5B2%5D=-1,x%5B1%5D=-6 (these are the coordinates of given points)


m=+13%2F5 Subtract the terms in the numerator 8--5 to get 13. Subtract the terms in the denominator -1--6 to get 5



So the slope is

m=13%2F5





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Now let's use the point-slope formula to find the equation of the line:




------Point-Slope Formula------
y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and (x%5B1%5D,y%5B1%5D) is one of the given points


So lets use the Point-Slope Formula to find the equation of the line


y--5=%2813%2F5%29%28x--6%29 Plug in m=13%2F5, x%5B1%5D=-6, and y%5B1%5D=-5 (these values are given)



y%2B5=%2813%2F5%29%28x--6%29 Rewrite y--5 as y%2B5



y%2B5=%2813%2F5%29%28x%2B6%29 Rewrite x--6 as x%2B6



y%2B5=%2813%2F5%29x%2B%2813%2F5%29%286%29 Distribute 13%2F5


y%2B5=%2813%2F5%29x%2B78%2F5 Multiply 13%2F5 and 6 to get 78%2F5

y=%2813%2F5%29x%2B78%2F5-5 Subtract 5 from both sides to isolate y


y=%2813%2F5%29x%2B53%2F5 Combine like terms 78%2F5 and -5 to get 53%2F5 (note: if you need help with combining fractions, check out this solver)



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Answer:



So the equation of the line which goes through the points (-6,-5) and (-1,8) is:y=%2813%2F5%29x%2B53%2F5


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=13%2F5 and the y-intercept is b=53%2F5


Notice if we graph the equation y=%2813%2F5%29x%2B53%2F5 and plot the points (-6,-5) and (-1,8), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=%2813%2F5%29x%2B53%2F5 through the points (-6,-5) and (-1,8)


Notice how the two points lie on the line. This graphically verifies our answer.