Seven Steps to Enlightenment...
Or Y= MX + B
Is Mr. Brügge's tattoo real?
The ability to find the equation for a line is a foundational concept in science. The graph above, although silly, will be used for "The Seven Steps to Enlightenment." It is assumed that the student has received some instruction in algebra to this point; this page is designed to be a reminder.
1. The first step is to pick two points on the line. Please note that not all data line up as nicely as this example. Unless a data point falls on the best-fit line, it should not be used.
Let's pick (1, 2) and (4, 8). Remember that the first in the pair is the X and the second is the Y. The horizontal (X) comes before the vertical (Y).
2. Now it's time to take the two points and write two equations. This is just a matter of inserting X and Y into our tattoo-worthy equation, Y = MX + B.
1st. 2 = 1M + B (Note that I've placed the value for X--which is 1--in front of the M as is the convention in algebra. A value of 1 does not really need to be written but for this example I want you to see how it works.)
2nd. 8 = 4M + B
3. Take the 1st equation, 2 = 1M + B, and solve for B. This is the fancy math way of saying that B needs to be an orphan and alone on one side.
There are a number of ways of doing this. My foolish example is to think of 1M hopping over the equals sign and magically becoming negative; you can also think of adding a negative 1M to both sides.
So now the equation looks like this: 2 -1M = B
4. Substitution time! Insert what B is equal to in the first equation into the second equation.
8 = 4M + B can also be written 8 = 4M + 2 -1M. It's a bit like taking four quarters and substituting a dollar bill. We've just rewritten the equation so that B has been replaced with M.
5. Now it's time to "clean up your room." Let's take the new equation and solve for M. This is just a matter of putting like things together.
8 = 4M + 2 -1M
Let's add the M's together: 8 = 3M + 2 (A positive 4M plus a negative 1M equals 3M)
Now it's time for the 2 to "hop" across the equals sign and become negative.
8 = 3M + 2 and after a "hop" we have 6 = 3M
Now it's the "ice cream" rule--whatever you do to one side of the equation, you must do the same thing to the other side. In this case we are going to divide each side by 3 to get M alone on one side.
6 = 3M so let's be fair 6/3 = 3/3M
After all this work, the clouds part, angels sing and we now have an actual number for M!
M = 2 This is known in the math world as the slope of the line.
6. Now that we know M as a number, we can substitute it back into our first equation.
2 = 1M + B
Substitute: 2 = (1)(2) + B
Simplify: 2 = 2 + B
Hop: 0 = B or B = 0 (More clouds parting and angels singing!)
7. We can now take the actual numbers for M and B and plug them into our general equation, Y= MX + B.
Y = 2X + 0
This equation, Y = 2X + 0, will now let us solve for ANY value of X. Let's say, for example, I hit my brother 100 times--this is X--I can then find out that I'll have 200 mL of blood. A person does not have to test every possible circumstance; instead, math can find the intermediate values
This is a very powerful tool!