# Intermediate Algebra/Slope

To find the slope of a line you have to use the graph:
With two points (x1,y1) and (x2,y2),

${\displaystyle m=(y2-y1)/(x2-x1)}$

An example of this is to find the slope of the two points (4,3) and (8,5).

${\displaystyle m=y2(5)-y1(3)/x2(8)-x1(4)}$
${\displaystyle m=5-3/8-4}$
${\displaystyle m=2/4}$
${\displaystyle m=1/2}$

Therefore the slope of those two points is 1/2. Now to graph the slope of a line you use the formula:

${\displaystyle y=mx+b}$

Where b = the Y-Intercept and m = the slope of the line.

But first we must find what the Y-Intercept is, to do this we choose one of the points that we are given, and input it into the formula. For this example, I'm going to use the point (4,3):

${\displaystyle y(3)=m(1/2)x(4)+b}$
${\displaystyle 3=1/2(4)+b}$
${\displaystyle 3(+2)=2(-2)+b}$
${\displaystyle 5=B}$

We now have the Y-Intercept, this can then be input into the formula to find the various points to graph. Using a simple T-Chart, pick various numbers and substitute them in for X, and you will find Y. Write these down, and graph them.

x = 0
${\displaystyle y=m(1/2)x(0)+5}$
${\displaystyle y=1/2(0)+5}$
${\displaystyle y=5}$

x = -2
${\displaystyle y=m(1/2)x(-2)+5}$
${\displaystyle y=1/2(-2)+5}$
${\displaystyle y=-1+5}$
${\displaystyle y=4}$

x = 2
${\displaystyle y=m(1/2)x(2)+5}$
${\displaystyle y=1/2(2)+5}$
${\displaystyle y=1+5}$
${\displaystyle y=6}$

X - Y
0 - 5
-2 - 4
2 - 6

Plot the points, and connect the dots to form your line.

Now a shortcut to doing this is to use the rise over run (Rise/Run) formula. Which is simply that whatever the slope is, (In the case of our example, 1/2) is the rise and run of our slope.

The rise is how many up from our Y-Intercept we go before we plot our point. The run is how many left or right (X-Axis) we move before we plot our point. So if we have a slope of 1/2, our Rise is 1 and our Run is 2. This means that we move 1 (one) up from our Y-Intercept and 2 (two) to the right (since 2 is a positive number, if it were -2 we would move to the left on the X-Axis). So our plot would be at (6,2), and the next point after that would be (7,4) and so on.

Rise/Run format is much easier and faster, but you should still understand how to use a T-Chart.