1. Slopes and Lines
What is slope?
 It is the incline at which a line connects from point A to point B
Let’s find slope in this line equation:
Ex: y =(1/3)x+5
We know that the equation of a line is y=mx+b, where m is the slope of a line. The
slope of the line is therefore 1/3, or m=(1/3). The slope can also be understood by
looking at a graph.
Look at point (-3,4). To get to point (0,5) the line moves up 1 and then to the right
3, and this can be seen in the dashes. This is the rise over the run, where 1 is the
rise and 3 is the run. The rise over the run can be rewritten as 1/3. The rise over
the run is also the slope, and so using this graph method the slope is 1/3.
How can I find slopes between two points?
Example: (3, -5), (-2, 1)
We know that to find the slope we will use the equation (y2-y1)/(x2-x1). We will
make (3, -5) point 1 which is (x1, y1) and (-2, 1) will be point 2 which is (x2, y2). Let’s
find the slope with the equation that was mentioned before.

2. 𝑦2−𝑦1
𝑥2−𝑥1
=
1−(−5)
(−2)−3
=
1+5
−2−3
=
6
−5
=
−6
5
= m = slope
The slope for these two points can also be found by reversing the points. Let’s
make (3, -5) point 2 and (-2, 1) point 1. Solve for the slope.
𝑦2−𝑦1
𝑥2−𝑥1
=
(−5)−1
3−(−2)
=
−5−1
3+2
=
−6
5
= m = slope
As you can see it doesn’t matter which point is chosen as the first or second, the
slope should end up the same for both.
How can I create a line after finding the slope?
Using the same example as the one before, we have found from the points (3, -5)
and (-2, 1) that the slope between them is -6/5. We can now use one of the points
and the slope to create a line equation. We know that the equation for a line is
y=mx+b. Let’s take point (3, -5) to figure out a line equation; it should be noted
that you could also use point (-2, 1) and find the same equation. Since (3, -5) is
simply (x, y), we can plug in 3 for x and -5 for y. We also know that m is -6/5 so we
can also plug that into the equation as well. All we do then is solve for b, which is
known as the y-intercept of the line.
y = mx + b
-5 = (-6/5) * (3) + b
-5 = (-18/5) + b
-5 – (-18/5) = b
(-25/5) + (18/5) = b
(-25 +18)/5 = b
-7/5 = b
We now know that b is equal to -7/5. Using this information we can now write a line
equation.
y = (-6/5)x + (-7/5)
If we plug in -2 for x then we find y to be 1, which is the point (-2,1) and proves
that the line runs through points (3, -5) and (-2, 1).
Here 3 is 3/1 and multiplied
by -6/5 we get -18/5.
Here -25/5 is equal to -5. We
make -5 a fraction to add
18/5 to it.