Slope and How to Graph Lines (it's really useful in business charts and stuff like that)

    Hello, again :). I hope you had a nice Valentines Day. In this post, I am going to explain the concept of slope and how to graph a linear equation. Now slope is basically the steepness of a line. The slope formula is actually pretty intimidating if you're not familiar with it, but don't panic. Here it is:

(y1-y2)/ (x1-x2)= m  

NOTE: (the little numbers next to the y's and x's aren't anything mathematical, it's just the first and second y's and x's that you come to. Imagine that you have the set of ordered pairs, that's just another way of saying the coordinates of some points on a graph, like (4,7) and (5,6). The 4 and the 5 would be your x's so you could label 4 x1 the 5 x2 {ordered pairs are always in the form (x,y) so your first number in the pair will always be your x and the second number will always be your y.} and your 7 and your 6 will be your y's so you can label the 7 y1 and the 6 y2.  After you have your numbers labeled, just plug them into the formula.)
    If you read the note above, you'll see how to label the ordered pairs (the sets of numbers hat you use to find slope), so let's do that now: say you have the two ordered pairs (9,2) and (5,8). Now, your x's are your 9 and your 5 you can label them; and since you know that your y's are your 2 and your 8, you can label them too. You should have x1 as 9 and x2 as 5. You should also have y1 as 2 and y2 as 8.

(Notice that the numbers from the same ordered pair have the same numbers in their label- 9 and 2 are from the same pair and they both have a 1 in their label, and the same of 5 and 8.)

Now that you have your values assigned, plug them into your formula and simplify:

(2-8)/(9-5)= m

Note: (a lower-case m stands for slope- DO NOT use an upper-case M- it means something completely different, but we'll get to that later).

Now, 2-8 is -6, and 9-5 is 4, so you should have -6 over 4 which will simplify to -3 over 2. (I'll get to simplifying fractions next post, but basically you just divide the top and the bottom number by the same number.)

In the end, your slope will be -3/2. However, to be able to use this slope effectively, you have to have a starting point on your graph; this is where the y-intercept formula comes in. The y-intercept formula looks like this: 

y=mx+b   the b stands for the y-intercept which is where the line you graph crosses the y- axis (it's the vertical line that makes up the coordinate plane). Once you know this value, you can use that as your starting point on your graph and use your slope from there.
    
Note:  (your slope is rise over run. If your slope was, for example, 3/5 or 3 over 5, and you already had a point on the graph, you would go up- or rise- 3 and go over- run- 5 from that point and plot another point where you stop after all the rising and running)

Now you know, that the lower-case m means slope, so to write out your formula, you plug in your slope into where the m is and solve for b. You should have something like this (given that you're using the same slope from above- it was -3/2 by the way):

y=(-3/2)x+b

Now, from earlier, you know that an ordered pair has numbers that can be labeled with an x and a y, so pick one of the points that you found your slope with (let's use (9,2)) and plug in the x and the y (the 9 will be your x and the 2 will be your y {remember (x,y)}). Once you plug everything in, you should have something like this:

2=(-3/2)(9)+b

Now simplify and solve for b:

2=(-27/2)+b
(I was mistaken when I said these would be easier numbers, but they are workable withable :)... I'm gonna change the 2 to 4 over 2. Then we can subtract the -27 over 2 without it getting messy- and, just FYI, 4 over 2 and 2 are the same thing, but, again, I'll get to that next post)

(4/2)= (-27/2)+b
add (-27/2) to both sides
(31/2)=b  or    15.5=b

With that, you now know that at 15.5 on the up-and-down axis is a point on your line. Now you can use the rise-over-run tidbit that I talked about in the last 'note' to plot your next point. Since your slope is negative, you will first rise 3 (go UP 3 from your point on the y-axis) then go LEFT 2. (you go left, because left is the negative direction and your slope is negative.) you should have a point at... (-2,18.5), and you always go in the left or right direction first then you go either up or down. So... you'll go to the left 2 then up 18.5 and that should be your other point. You'll then draw your line through your point at 15.5 and through (-2,18.5) and you'll have a line.

Alright piece peace out www.khanacademy.org or more help

-bye bye! :D