To find the equation of a line in slope-intercept form given two points, we must arrange the points this way to find the slope (
![m](https://tex.z-dn.net/?f=m)
):
![m = \frac{y_{2}-y _{1}}{x_{2}-x_{1}}](https://tex.z-dn.net/?f=m%20%3D%20%20%5Cfrac%7By_%7B2%7D-y%20_%7B1%7D%7D%7Bx_%7B2%7D-x_%7B1%7D%7D%20)
We can replace the variables in this equation with the values from the points we have been given, (
![\frac{-1}{2}, \frac{-7}{2}](https://tex.z-dn.net/?f=%20%5Cfrac%7B-1%7D%7B2%7D%2C%20%20%5Cfrac%7B-7%7D%7B2%7D%20)
) and (2, 14). We know that the x-values always come first in an ordered pair, so we can put those in our equation first.
It doesn't really matter which x-value goes in which x-value slot in the equation as long as you match up the y-values in the same fashion. But for the sake of convenience, we will call the 2 in the second ordered pair
![x_{2}](https://tex.z-dn.net/?f=%20x_%7B2%7D%20)
and the
![\frac{-1}{2}](https://tex.z-dn.net/?f=%20%5Cfrac%7B-1%7D%7B2%7D%20)
in the first ordered pair
![x_{1}](https://tex.z-dn.net/?f=%20x_%7B1%7D%20)
.
Now, we can match these values in our equation, and while we're at it, we can substitute the appropriate y-values into their places as well.
![m = \frac{y_{2}-y _{1}}{x_{2}-x_{1}}](https://tex.z-dn.net/?f=m%20%3D%20%20%5Cfrac%7By_%7B2%7D-y%20_%7B1%7D%7D%7Bx_%7B2%7D-x_%7B1%7D%7D%20)
![m = \frac{14 - \frac{-7}{2} }{2 - \frac{-1}{2} }](https://tex.z-dn.net/?f=m%20%3D%20%20%5Cfrac%7B14%20-%20%20%5Cfrac%7B-7%7D%7B2%7D%20%7D%7B2%20-%20%20%5Cfrac%7B-1%7D%7B2%7D%20%7D%20)
Now, we can see that we are subtracting negatives in this equation. Remember that whenever you subtract a negative, it is the same as adding a positive. So, this equation could be rewritten as
![m = \frac{14 + \frac{7}{2} }{2 + \frac{1}{2} }](https://tex.z-dn.net/?f=m%20%3D%20%20%5Cfrac%7B14%20%2B%20%20%5Cfrac%7B7%7D%7B2%7D%20%7D%7B2%20%2B%20%20%5Cfrac%7B1%7D%7B2%7D%20%7D%20)
and we can change the improper fraction in the numerator into a mixed number for ease of addition.
![m = \frac{14 + 3 \frac{1}{2} }{2+ \frac{1}{2} }](https://tex.z-dn.net/?f=m%20%3D%20%20%5Cfrac%7B14%20%2B%203%20%5Cfrac%7B1%7D%7B2%7D%20%7D%7B2%2B%20%5Cfrac%7B1%7D%7B2%7D%20%7D%20)
And here, we can add, since it's made simple for us.
![m = \frac{17 \frac{1}{2} }{2 \frac{1}{2} }](https://tex.z-dn.net/?f=m%20%3D%20%20%5Cfrac%7B17%20%20%5Cfrac%7B1%7D%7B2%7D%20%7D%7B2%20%20%5Cfrac%7B1%7D%7B2%7D%20%7D%20)
Finally, to get the slope we can complete the fraction by dividing.
![17.5 ÷ 2.5 = 7](https://tex.z-dn.net/?f=17.5%20%C3%B7%202.5%20%3D%207)
The slope of this equation,
![m](https://tex.z-dn.net/?f=m)
, is 7, and so far, our equation looks like this:
![y = 7m + b](https://tex.z-dn.net/?f=y%20%3D%207m%20%2B%20b)
Now, to find y-intercept.
To do this, we simply have to substitute one of the ordered pairs in for the appropriate x- and y-values and solve. To make it easier on ourselves, we can use (2, 14) so we don't have to deal with negative numbers or fractions.
![y = 7x + b](https://tex.z-dn.net/?f=y%20%3D%207x%20%2B%20b)
Let us substitute in our values.
![14=7(2)+b](https://tex.z-dn.net/?f=14%3D7%282%29%2Bb)
Now, we can solve by multiplying the 7 and 2.
![14=14+b](https://tex.z-dn.net/?f=14%3D14%2Bb)
Here, we can subtract 14 from both sides, since it is added to both in the equation, and we are left with
![0 = b](https://tex.z-dn.net/?f=0%20%3D%20b)
which can be flipped around to show
![b=0](https://tex.z-dn.net/?f=b%3D0)
The y-intercept of this line is 0.
Now, we can make this known in our equation like this:
![y=7x+0](https://tex.z-dn.net/?f=y%3D7x%2B0)
Or, for neatness' sake, we can just say
![y=7x](https://tex.z-dn.net/?f=y%3D7x)
which is your final equation.
The line through (
) and (2, 14) in slope-intercept form is <span>
.
Hope that helped! =)
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