SOLUTION: write an equation of this line using slope intercept form: a.) passing through the points (-2,3)(1,3) b.) passing through the point(3,1) and parallel to the line y=1/3x-2 c.) pa

Algebra ->  Equations -> SOLUTION: write an equation of this line using slope intercept form: a.) passing through the points (-2,3)(1,3) b.) passing through the point(3,1) and parallel to the line y=1/3x-2 c.) pa      Log On


   

Question 117626: write an equation of this line using slope intercept form:
a.) passing through the points (-2,3)(1,3)
b.) passing through the point(3,1) and parallel to the line y=1/3x-2
c.) passing through the point (3,1)and perpendicular to the line y=1/3x-2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a)

First lets find the slope through the points (-2,3) and (1,3)

m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: is the first point (-2,3) and is the second point (1,3))

m=%283-3%29%2F%281--2%29 Plug in y%5B2%5D=3,y%5B1%5D=3,x%5B2%5D=1,x%5B1%5D=-2 (these are the coordinates of given points)

m=+0%2F3 Subtract the terms in the numerator 3-3 to get 0. Subtract the terms in the denominator 1--2 to get 3


m=0 Reduce

So the slope is
m=0

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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 is one of the given points

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

y-3=%280%29%28x--2%29 Plug in m=0, x%5B1%5D=-2, and y%5B1%5D=3 (these values are given)


y-3=%280%29%28x%2B2%29 Rewrite x--2 as x%2B2


y-3=0x%2B%280%29%282%29 Distribute 0

y-3=0x%2B0 Multiply 0 and 2 to get 0%2F0. Now reduce 0%2F0 to get 0

y=0x%2B0%2B3 Add 3 to both sides to isolate y

y=0x%2B3 Combine like terms 0 and 3 to get 3


y=3 Remove the zero term
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Answer:


So the equation of the line which goes through the points (-2,3) and (1,3) is: y=3

The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=0 and the y-intercept is b=3

Notice if we graph the equation y=3 and plot the points (-2,3) and (1,3), we get this: (note: if you need help with graphing, check out this solver)

Graph of y=3 through the points (-2,3) and (1,3)

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




b)


Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line


Since any two parallel lines have the same slope we know the slope of the unknown line is 1%2F3 (its from the slope of y=%281%2F3%29%2Ax-2 which is also 1%2F3). Also since the unknown line goes through (3,1), we can find the equation by plugging in this info into the point-slope formula

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 the given point



y-1=%281%2F3%29%2A%28x-3%29 Plug in m=1%2F3, x%5B1%5D=3, and y%5B1%5D=1



y-1=%281%2F3%29%2Ax-%281%2F3%29%283%29 Distribute 1%2F3



y-1=%281%2F3%29%2Ax-3%2F3 Multiply



y=%281%2F3%29%2Ax-3%2F3%2B1Add 1 to both sides to isolate y

y=%281%2F3%29%2Ax-3%2F3%2B3%2F3 Make into equivalent fractions with equal denominators



y=%281%2F3%29%2Ax%2B0%2F3 Combine the fractions



y=%281%2F3%29%2Ax%2B0 Reduce any fractions

So the equation of the line that is parallel to y=%281%2F3%29%2Ax-2 and goes through (3,1) is y=%281%2F3%29%2Ax%2B0


So here are the graphs of the equations y=%281%2F3%29%2Ax-2 and y=%281%2F3%29%2Ax%2B0



graph of the given equation y=%281%2F3%29%2Ax-2 (red) and graph of the line y=%281%2F3%29%2Ax%2B0(green) that is parallel to the given graph and goes through (3,1)









c)


Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line


Remember, any two perpendicular lines are negative reciprocals of each other. So if you're given the slope of 1%2F3, you can find the perpendicular slope by this formula:

m%5Bp%5D=-1%2Fm where m%5Bp%5D is the perpendicular slope


m%5Bp%5D=-1%2F%281%2F3%29 So plug in the given slope to find the perpendicular slope



m%5Bp%5D=%28-1%2F1%29%283%2F1%29 When you divide fractions, you multiply the first fraction (which is really 1%2F1) by the reciprocal of the second



m%5Bp%5D=-3%2F1 Multiply the fractions.


So the perpendicular slope is -3



So now we know the slope of the unknown line is -3 (its the negative reciprocal of 1%2F3 from the line y=%281%2F3%29%2Ax-2). Also since the unknown line goes through (3,1), we can find the equation by plugging in this info into the point-slope formula

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 the given point



y-1=-3%2A%28x-3%29 Plug in m=-3, x%5B1%5D=3, and y%5B1%5D=1



y-1=-3%2Ax%2B%283%29%283%29 Distribute -3



y-1=-3%2Ax%2B9 Multiply



y=-3%2Ax%2B9%2B1Add 1 to both sides to isolate y

y=-3%2Ax%2B10 Combine like terms

So the equation of the line that is perpendicular to y=%281%2F3%29%2Ax-2 and goes through (3,1) is y=-3%2Ax%2B10


So here are the graphs of the equations y=%281%2F3%29%2Ax-2 and y=-3%2Ax%2B10




graph of the given equation y=%281%2F3%29%2Ax-2 (red) and graph of the line y=-3%2Ax%2B10(green) that is perpendicular to the given graph and goes through (3,1)