SOLUTION: How do you write an equation for the line (in slope-intercept form) with the given information: 1.is vertical and passes through (4, 11) 2.passs thrugh (7, -10) and (4, -16) 3.p

Algebra ->  Expressions-with-variables -> SOLUTION: How do you write an equation for the line (in slope-intercept form) with the given information: 1.is vertical and passes through (4, 11) 2.passs thrugh (7, -10) and (4, -16) 3.p      Log On


   

Question 118677: How do you write an equation for the line (in slope-intercept form) with the given information:
1.is vertical and passes through (4, 11)
2.passs thrugh (7, -10) and (4, -16)
3.passes through (2, 18) and (4, 17)

(P.s. they are seperate problems 123)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
#1
All vertical lines are of the form x=k where k is the x-coordinate of the points that the line goes through. So in this case, the equation of the line that passes through (4, 11) is x=4

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#2

First lets find the slope through the points (7,-10) and (4,-16)

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 (7,-10) and is the second point (4,-16))

m=%28-16--10%29%2F%284-7%29 Plug in y%5B2%5D=-16,y%5B1%5D=-10,x%5B2%5D=4,x%5B1%5D=7 (these are the coordinates of given points)

m=+-6%2F-3 Subtract the terms in the numerator -16--10 to get -6. Subtract the terms in the denominator 4-7 to get -3


m=2 Reduce

So the slope is
m=2

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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--10=%282%29%28x-7%29 Plug in m=2, x%5B1%5D=7, and y%5B1%5D=-10 (these values are given)


y%2B10=%282%29%28x-7%29 Rewrite y--10 as y%2B10


y%2B10=2x%2B%282%29%28-7%29 Distribute 2

y%2B10=2x-14 Multiply 2 and -7 to get -14

y=2x-14-10 Subtract 10 from both sides to isolate y

y=2x-24 Combine like terms -14 and -10 to get -24
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Answer:


So the equation of the line which goes through the points (7,-10) and (4,-16) is:y=2x-24

The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=2 and the y-intercept is b=-24

Notice if we graph the equation y=2x-24 and plot the points (7,-10) and (4,-16), we get this: (note: if you need help with graphing, check out this solver)

Graph of y=2x-24 through the points (7,-10) and (4,-16)

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


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#3

First lets find the slope through the points (2,18) and (4,17)

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 (2,18) and is the second point (4,17))

m=%2817-18%29%2F%284-2%29 Plug in y%5B2%5D=17,y%5B1%5D=18,x%5B2%5D=4,x%5B1%5D=2 (these are the coordinates of given points)

m=+-1%2F2 Subtract the terms in the numerator 17-18 to get -1. Subtract the terms in the denominator 4-2 to get 2

So the slope is
m=-1%2F2

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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-18=%28-1%2F2%29%28x-2%29 Plug in m=-1%2F2, x%5B1%5D=2, and y%5B1%5D=18 (these values are given)


y-18=%28-1%2F2%29x%2B%28-1%2F2%29%28-2%29 Distribute -1%2F2

y-18=%28-1%2F2%29x%2B1 Multiply -1%2F2 and -2 to get 2%2F2. Now reduce 2%2F2 to get 1

y=%28-1%2F2%29x%2B1%2B18 Add 18 to both sides to isolate y

y=%28-1%2F2%29x%2B19 Combine like terms 1 and 18 to get 19
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Answer:


So the equation of the line which goes through the points (2,18) and (4,17) is:y=%28-1%2F2%29x%2B19

The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=-1%2F2 and the y-intercept is b=19

Notice if we graph the equation y=%28-1%2F2%29x%2B19 and plot the points (2,18) and (4,17), we get this: (note: if you need help with graphing, check out this solver)

Graph of y=%28-1%2F2%29x%2B19 through the points (2,18) and (4,17)

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