What is the equation of the line that passes through (-8,9) and (10,-3)?
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I am confused how to find "b". The answer is in point-slope form but i don't know how to get there. HELP!!! thx
Other Answers (6)Rated Highest
By letting p1=( -8, 9 ) and p2 = ( 10, -3 ), we have the slope of
m = change y / change x = -2/3
Using p1 ( -8, 9 ) in the point-slope formula, we have:
( y - 9 ) = -2/3 ( x - 8 )
solving for y = -2/3 x + 11/3 ( y - intercept is b = 11/3 when x =0 )
Using p2 ( 10, -3 ) in the point-slope formula yields us once again....
( y + 3 ) = -2/3 ( x - 10 ) expanding and collecting the like terms we obtain
y = -2/3 x + 11/3 ( y-intercept b = 11/3 when x =0 )
multiplying both sides of the equation by 3, we have:
3y = -2x + 11
therefore, the equation of the line that passes through the
points ( -8, 9 ) and ( 10, -3 ) is:
2x +3y - 11 = 0
Let me give you TWO methods, in detail:
The equation is y = m x + b
The line passes through (-8,9) and (10,-3) when both of the following equations hold:
9 = -8 m + b
-3 = 10 m + b
Substract those two equations from each other to eliminate b. You obtain: 12 = -18 m, or m = -2/3.
Plug that value of m into either equation to obtain the value of b, namely (using the first equation): 9 = -8 (-2/3) + b, so:
b = 9-16/3 = 11/3
If you prefer, you can also obtain the value of b directly by eliminating m between the two equations (multiply the first by 5 and the second by 4 and add them up). You obtain: 45-12 = 9 b, so b = 11/3, indeed.
SECOND METHOD (more efficient):
At a slightly more advanced level, we may obtain the equation of the line directly by stating that (x+8, y-9) is proportional to (10+8,-3-9). If you know what a determinant is, you may just write that the relevant 2 by 2 determinant is zero. Otherwise, just write that the proportionality means:
(x+8)(-3-9) = (y-9)(10+8)
Just simplify that:
-12 (x+8) = 18 (y-9) or 2(x+8)+3(y-9) = 0
This boils down to 2x+3y=11
You may keep this equation as is or divide by 3 to put it in the form
y = (-2/3)x+(11/3) or y = (11-2x) / 3
I advise you to get familiar with the second method ASAP and, also, to acquire a taste for the nicer form of the equation of a line where the coefficient of y need not be 1. Here: 2x+3y=11.
Last step: Check that the equation obtained holds for both points:
2(-8) + 3(9) = 11
2(10) + 3(-3) = 11
Trimming all the fat, the entire computation merely consists in simplifying a single line, as shown in the link below:
first find the slope: rise over run y2-y1/ x2-x1
y-9= -2/3 ( x- (-8) )
y-9= -2/3x -16/3
add 9 to get y by itself
y= -2/3x + 11/3
First, you have to figure out what the slope is. The formula for doing that is this: y2-y1/x2-x1 = the slope of the line. OK?
So, lets use the first point as (x1, y1) and the second point as (x2, y2).
Considering that, we'd have -3-9/10-(-8) = -12/18 = -2/3 = slope
Now, use the point/slope forumula to get the equation you need.
Here it is: y-y1 = m(x-x1)
The first y and x stay just that way, as y and x. The y1 and x1 are the coordinates of whichever point given you choose to work with. I'll choose point #2.
y - 9 = m(x - (-8))
y - 9 = -2/3(x + 8)
now multiply the right side out:
y - 9 = -2/3 x - 16/3
The form you want to put this in is y = mx + b, I know this because you mentioned b in your question. b indicates the y intercept of the line between these 2 points.
So, now you have to add 9 to both sides of the equation above to get it in the form of y = mx + b:
y -9 +9 = -2/3 x - 16/3 + 9
y = -2/3 x - 16/3 + 27/3
y = -2/3 x + 27/3 - 16/3
y = -2/3 x + 11/3 Done!
Source(s):BS math, teacher
First find m, or the slope.
m = (y2 - y1) / (x2 - x1)
m = (-3 - 9) / (10 - (-8) )
m = -12 / 18
m = -2/3
Point slope form:
(y - y1) = m (x - x1)
Pick an ordered pair. I'll pick (-8, 9)
(y - 9) = (-2/3) (x - -8)
(y - 9) = (-2/3) (x + 8)
y - 9 = (-2/3)x - 16/3
y = (-2/3)x - 16/3 + 9
y = (-2/3)x + 11/3
First find the slope 'm'
m =( y" -y') /( x"-x') = -3-9 / 10-(-8) = -12 / 18 = -2/3
so the equation in slope-intercept form will be
y = -2x/3 + b
now substitute x=-8 and y=9
9 = -2x-8 /3 +b
9 = 16/3 +b
b = 9-16/3
b = (27-16)/3
b = 11/3
so the equation of the line is y = -2x/3 +11/3
or , 3y +2x = 11
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