write an equation of a line given two points?

Question

the problem is

(2,-1) (5,3)

i need to know the steps!

help!

Answer 1

(i apoligize 4 the signs, i tried 2 do it as neat as possible)
First of all, remember what the equation of a line is:

y = mx+b

Where: m is the slope, and b is the y-intercept.

First, let's find what m is, the slope of the line...

The slope of a line is a measure of how fast the line "goes up" or "goes down". A large slope means the line goes up or down really fast (a very steep line). Small slopes means the line isn't very steep. A slope of zero means the line has no steepness at all; it is perfectly horizontal.
For lines like these, the slope is always defined as "the change in y over the change in x" or, in equation form:

m= (Y2-Y1)
----------
(X2-X1)


So what we need now are the two points you gave that the line passes through. Let's call the first point you gave, (2,-1), point #1, so the x and y numbers given will be called x1 and y1. Or, x1=2 and y1=-1.
Also, let's call the second point you gave, (5,3), point #2, so the x and y numbers here will be called x2 and y2. Or, x2=5 and y2=3.


Now, just plug the numbers into the formula for m above, like this:


m= 3 - -1
--------
5 - 2



or...


m= 4
----
3



or...


m=4/3

So, we have the first piece to finding the equation of this line, and we can fill it into y=mx+b like this:


y=4/3x+b

Now, what about b, the y-intercept?

To find b, think about what your (x,y) points mean:
(2,-1). When x of the line is 2, y of the line must be -1.
(5,3). When x of the line is 5, y of the line must be 3.
Because you said the line passes through each one of these two points, right?
Now, look at our line's equation so far: y=4/3x+b. b is what we want, the 4/3 is already set and x and y are just two "free variables" sitting there. We can plug anything we want in for x and y here, but we want the equation for the line that specfically passes through the two points (2,-1) and (5,3).

So, why not plug in for x and y from one of our (x,y) points that we know the line passes through? This will allow us to solve for b for the particular line that passes through the two points you gave!.

You can use either (x,y) point you want..the answer will be the same:


(2,-1). y=mx+b or -1=4/3 × 2+b, or solving for b: b=-1-(4/3)(2). b=-11/3.

(5,3). y=mx+b or 3=4/3 × 5+b, or solving for b: b=3-(4/3)(5). b=-11/3.
See! In both cases we got the same value for b. And this completes our problem.

The equation of the line that passes through the points
(2,-1) and (5,3)

is

y=(4/3)x-(11/3)

or

y= 4 11
-- x - ---
3 3

Answer 2

First, find the slope by subtracting the y's divided by the subtraction of the x's.
3-(-1)/5-2)= 4/3 slope=m
Now, use y=mx+b to fill in m and one of the ordered pairs.
If you would use (2,-1), then -1= 4/3(2)+b
-1= 8/3+b
-1+-2 2/3 = b
-3 2/3=b this is the y intercept
Now, put the slope and y intercept in y=mx+b
y=4/3x+ (-3 2/3)

Answer 3

m = y2-y1/x2-x1
= -1-3/2-5
= -4/-3
= 4/3

Therefore, the slope is 4/3.

Since y=mx+b, we now know that y=4/3x+b

Substitute one of the coordinates --> (5,3)

3 = 4/3 (5) +b
3 = 20/3 + b
b = 3 - 20/3
= -11/3

Therefore, the equation is y=4/3x-11/3

Answer 4

The points must be different in order to determine the slope of the line. If they were both the same point then you would have no idea what the slope is and thus unable to determine the equation of a line. The slope equation is m = (y2 - y1)/(x2 - x1)

Answer 5

first do y2-y1 divided by x2-x1 so 3-(-1) over 5-2. once you find that number, make sure it simplified and that will be your slope(m) or the number that comes before x in slope intercept form. crap i forgot which coordinate you use for the y intercept. well you kno slope-intercept form is y=mx+b. (b being the y-intercept) hope this helps!

Answer 6

for: y = ax + b

-1-3 = -4
2-5 = -3
-4/-3 = 4/3 = a

3 = 5 (4/3) + b
3 = 20/3 + b
3 - (20/3) = b
-11/3 = b

so: y = (4/3)x - (11/3)

Answer 7

First you find the slope, m.
M=[3- (-1)] / [5-2] = 4/3
Now, because you have the slope and a point, use point-slope:
y+1 = (4/3)(x-2)
y = (4/3)x - 5/3

Answer 8

m = (3 + 1) / (5 - 2) = 4 / 3
y - 3 = (4/3) (x - 5)
y - 3 = (4/3)x - 20/3
y = (4/3)x - 20/3 + 9/3
y = (4/3)x - 11/3

Answer 9

First you calculate the slope:

m = (Y2-Y1)/(X2-X1)

Then, you replace one of the two points in
Y = m*X+b

And you isolate b.