Question

The line through (-1/2, -7/2) and (2, 14) in slope intercept form??

Answer 1

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 ($m$):

$m = \frac{y_{2}-y _{1}}{x_{2}-x_{1}}$

We can replace the variables in this equation with the values from the points we have been given, ($\frac{-1}{2}, \frac{-7}{2}$) 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 $x_{2}$ and the $\frac{-1}{2}$ in the first ordered pair $x_{1}$.
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.

$m = \frac{y_{2}-y _{1}}{x_{2}-x_{1}}$

$m = \frac{14 - \frac{-7}{2} }{2 - \frac{-1}{2} }$

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

$m = \frac{14 + \frac{7}{2} }{2 + \frac{1}{2} }$

and we can change the improper fraction in the numerator into a mixed number for ease of addition.

$m = \frac{14 + 3 \frac{1}{2} }{2+ \frac{1}{2} }$

And here, we can add, since it's made simple for us.

$m = \frac{17 \frac{1}{2} }{2 \frac{1}{2} }$

Finally, to get the slope we can complete the fraction by dividing.

$17.5 ÷ 2.5 = 7$

The slope of this equation, $m$, is 7, and so far, our equation looks like this:

$y = 7m + b$

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.

$y = 7x + b$

Let us substitute in our values.

$14=7(2)+b$

Now, we can solve by multiplying the 7 and 2.

$14=14+b$

Here, we can subtract 14 from both sides, since it is added to both in the equation, and we are left with

$0 = b$

which can be flipped around to show

$b=0$

The y-intercept of this line is 0.

Now, we can make this known in our equation like this:

$y=7x+0$

Or, for neatness' sake, we can just say

$y=7x$

which is your final equation.

The line through ($\frac{-1}{2}, \frac{-7}{2}$) and (2, 14) in slope-intercept form is $y=7x$.
Hope that helped! =)