SOLUTION: Please help me with these two problems. Find an equation of the line with the given slope and through the given point. 1. slope=2;(3,11) & Find

Algebra ->  Graphs -> SOLUTION: Please help me with these two problems. Find an equation of the line with the given slope and through the given point. 1. slope=2;(3,11) & Find      Log On


   

Question 98452: Please help me with these two problems.
Find an equation of the line with the given slope and through the given point.
1. slope=2;(3,11)
&
Find an equation of the line through the given points.
1.(1,-5);(4,7)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
1.

"Find an equation of the line with the given slope and through the given point.
1. slope=2;(3,11)"


If you want to find the equation of line with a given a slope of 2 which goes through the point (3,11), you can simply use the point-slope formula to find the equation:


---Point-Slope Formula---
y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and is the given point

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

y-11=%282%29%28x-3%29 Plug in m=2, x%5B1%5D=3, and y%5B1%5D=11 (these values are given)


y-11=2x%2B%282%29%28-3%29 Distribute 2

y-11=2x-6 Multiply 2 and -3 to get -6

y=2x-6%2B11 Add 11 to both sides to isolate y

y=2x%2B5 Combine like terms -6 and 11 to get 5
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Answer:


So the equation of the line with a slope of 2 which goes through the point (3,11) is:

y=2x%2B5 which is now in y=mx%2Bb form where the slope is m=2 and the y-intercept is b=5

Notice if we graph the equation y=2x%2B5 and plot the point (3,11), we get (note: if you need help with graphing, check out this solver)

Graph of y=2x%2B5 through the point (3,11)
and we can see that the point lies on the line. Since we know the equation has a slope of 2 and goes through the point (3,11), this verifies our answer.





2.

"Find an equation of the line through the given points.
(1,-5);(4,7)"


First lets find the slope through the points (1,-5) and (4,7)

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 (1,-5) and is the second point (4,7))

m=%287--5%29%2F%284-1%29 Plug in y%5B2%5D=7,y%5B1%5D=-5,x%5B2%5D=4,x%5B1%5D=1 (these are the coordinates of given points)

m=+12%2F3 Subtract the terms in the numerator 7--5 to get 12. Subtract the terms in the denominator 4-1 to get 3


m=4 Reduce

So the slope is
m=4

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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--5=%284%29%28x-1%29 Plug in m=4, x%5B1%5D=1, and y%5B1%5D=-5 (these values are given)


y%2B5=%284%29%28x-1%29 Rewrite y--5 as y%2B5


y%2B5=4x%2B%284%29%28-1%29 Distribute 4

y%2B5=4x-4 Multiply 4 and -1 to get -4

y=4x-4-5 Subtract 5 from both sides to isolate y

y=4x-9 Combine like terms -4 and -5 to get -9
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Answer:


So the equation of the line which goes through the points (1,-5) and (4,7) is:y=4x-9

The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=4 and the y-intercept is b=-9

Notice if we graph the equation y=4x-9 and plot the points (1,-5) and (4,7), we get this: (note: if you need help with graphing, check out this solver)

Graph of y=4x-9 through the points (1,-5) and (4,7)

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