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! =)
