College Algebra - Linear Equations
Let me start with some basics, and if this is too basic, just let me know.
So - the philosohy of linear equations. What is their raison d'etre, and why are you being made to learn them?
Think about it like this.
There is some relationship that exists in nature (or in life, if you prefer) between things you know (or can control), and things you'd like to know that depend on those 'controllable' things.
You want to get a hold of that 'relationship' and make it work for you - so you can describe and predict stuff 'in life' that follows that relationship.
That's what math can do for us!!
All that notation - the x's the y's, the operators - and all that graphing, is supposed to make it easier! And it does. But only if you speak the language and understand how to translate:
What you want to know to help you in life
to
Mathematical symbology
I am going to try to make this clear with an example and maybe we can work through some questions you have, or problems you're working on.
Example to follow in a few...sound good?
@CaloenasNicobarica
@BurkeDevlin
Interesting. It's all still kinda floating around my pigeony brain, yet to be absorbed...but I get the gist of it. Do continue! I managed to snag a few problems as well that relate to what is being such a bugaboo. BTW Thanks for starting this thread and all this! *sets some tea down on the table for you*
@CaloenasNicobarica My pleasure. *sip* Thanks!
@CaloenasNicobarica Something I'd like to point out about the previous example. Notice that I illustrated the exact same relationship between (number of hours of tutoring) and (total earned) in 3 different ways.
1. In words.
2. As a linear equation.
3. Graphically.
The idea here is that, from 1 -> 3, it should get easier for you to tell me the 'output' of the relationship (total $) given the 'input' (hours).
I'm kind of trying to get across the motivation for the symbols and graphs and the connections between those and what you really, actually want to know.
How badly am I failing to do so? 
I'm new at tutoring this way.
@BurkeDevlin
@BurkeDevlin
You're not failing at anything. I like your dorky graphy chartboo! So 5 hours = coffee. *exchanges out tea for a pot of coffee instead*
But seriously. I do see the correlation. And once you plug in the numbers it works as a machine. I mean- in programming this is basically what's happening. I can do equations like this, and I can see the graphical correlation... Hnn... There's this one mechanisms that appear to be the same that I'm struggling with. Oh wait... now that I'm looking at it...it actually makes sense. O___O; So the x is variable, and then y adjusts and those are also your coordinates. I see.
Alright. So... I have some example problems that I kinda wanna see worked out. For some reason it feels like I'm missing something- but I'm pretty sure it's the same procedure!
x-2y = 15
Determine the missing coordinate pair in the ordered pair (-1, ?) so that it will satisfy the given equation.
Oh, I can join you in there in a sec. TY.
@CaloenasNicobarica Sure! (And thanks for the good feedback. Any specific questions, just ask.)
x-2y = 15
Determine the missing coordinate pair in the ordered pair (-1, ?) so that it will satisfy the given equation.
The trick here seems to be that it is using the terminology of graphing solutions to refer to satisfying the equation.
You know that an ordered pair is of the form (x, y). So they are telling you that the 'x' is -1 and asking for the 'y'.
(-1) - 2y = 15
I want to know 'y' so I want to get it by itself.
I can add 1 to both sides of the equation and it will still be 'true'.
(-1) - 2y + 1 = 15 + 1
-2y = 16
I can divide both sides by (-2) and it will still be 'true'.
(-2y / -2) = (16 / -2)
y = -8
Let's go back to the original equation and double-check.
x - 2y = 15
(-1) - 2(-8) = 15
-1 + 16 = 15
Yes!
Does that help? If not, let's go back to whatever step you may have gotten lost on.
@CaloenasNicobarica I just popped into 'School and College Support' chatroom if it would help.
@CaloenasNicobarica
Graphing fractions - here is an ordered pair (1/4, 3/2). I've expanded the scale a little to show that there is a lot of room between the 0 and 1, and between the 1 and 2. Each axis is an entire continuum, not just a discrete set of whole numbers. Usually only the whole numbers are labeled, but any fractional number can fit in between.
@BurkeDevlin I am planning on applying for the Science Management course at UCL for which I need linear algebra. This sounds so useful
@Lisax Good luck with the program!
I'm really glad this is helpful to you. 'Linear algebra' goes a bit further than this stuff, but it's really cool and its applications are everywhere. Machine learning algorithms tend to make heavy use of it. I think you'll have fun with it.
Feel free to post if you have any questions and I'll answer when I can.