Calling all math geniuses
I am really stuck with Linear Equations, I do not get it, at all....
Deveno, I am no good at math, but what I saw on the board, was what linalg123 said, and other ones, of course...
So, it was something like... y = -3x + 1 idk something like that...
well, that is not a "linear equation", but rather "the equation of a line" (the difference may not seem that big a deal to you, but it is important later on).
as you may be aware, a line can be completely decided if we know two points that lie on it (imagine lining up an "infinite ruler" on the two points, and then running one's pencil alongside the edge of the ruler). another way of describing a line is to say it is a "curve" that is straight (so an "uncurvy" curve, sort of bizarre, hmm?), with a constant "tilt".
the technical term for "tilt" is "slope". the usual way to measure slope is:
rise/run (how far up, divided by how far over).
well, if our two points are (x_{1},y_{1}) and (x_{2},y_{2}) the "rise" is the difference of the y's (we usually draw the y-axis runnning "up and down"), and the "run" is the difference of the x's.
for example, let's say we have a line going through (3,3) and (4,4). then the difference of the y's is 1 (4 -3 = 1) and the difference of the x's is also 1 (4 - 3 = 1, again), so the slope is 1/1 = 1 (we go "one over, and one up").
similarly, a line that goes through (1,1) and (4,2) has a slope of 1/3 = (2-1)/(4 -1) (we are only rising 1 for every 3 we go over). this line isn't as "steep" as the first one (the slope is smaller).
but naming other points besides the two we know (the ones we drew the line from) seems quite complicated. the slope of a line seems like one good way to tell different lines apart, but what about two parallel lines? they will both have the same "steepness", but they will obviously go through "different points".
what we need, is some way of telling WHICH line with a given slope we have. the easiest way to do this, is to measure "how high the line is" when x = 0. often this is called the "slope-intercept form" of a line:
y = mx + b
where m is the slope, and b is the "y-intercept" (which is just a fancy name for the height of the line at x = 0).
let's see if this new-fangled way of looking at slope matches our old one.
suppose (x_{1},y_{1}) and (x_{2},y_{2}) are both on our line. this means:
y_{1} = mx_{1} + b and
y_{2} = mx_{2} + b
so our two points are (x_{1},mx_{1}+b) and (x_{2},mx_{2}+b).
the difference of the y's is: (mx_{2}+b) - (mx_{1}+b) = mx_{2} + b - mx_{1} - b = mx_{2} - mx_{1} = m(x_{2} - x_{1}).
the difference of the x's is: x_{2} - x_{1}.
so slope = rise/run = m(x_{2} - x_{1})/(x_{2} - x_{1}) = m. hey! it works.
let's look at your example:
y = -3x + 1
here, m = -3, and b = 1. this tells us the line has a slope of -3 (it goes down 3 as we go over (right) 1), and when x = 0, it is "1 high".
let's check this: suppose x = 0.
then y = -3(0) + 1 = 0 + 1 = 1, so the point on the y-axis that the line crosses is (0,1).
now if our slope is -3, we should get a new point also on our line if we go over to the right by 1, and down by 3.
well 1 over from 0 is 0+1 = 1, and 3 down from 1 is 1 - 3 = -2, so (1,-2) should be a point on our line.
so let's check, we set x = 1:
y = -3(1) + 1 = -3 + 1 = -2.
that's really all there is to it.
I asked my teacher to help me and I think I have a little bit of a better understanding of it... But I still have to get the full grasp of it.
I think it'd be better if I had a step by step talk through with you guys.
I small example in my book is...
Find the gradient of:
P(3,1), Q(7,3)
gradient = rise/run =$\displaystyle \frac{y_2-y_1}{x_2-x_1}$
So, if you have the points P(x_{1},y_{1}) and Q(x_{2},y_{2}), you should be able to work it out from there.
If you don't do well remembering formulas, just think of it logically. As x increases from 3 to 7, y increases from 1 to 3. So an increase of 4 in x leads to an increase of 2 in y.
so, all the question is really asking is, if x increases by ONE, how much does y increase by?
As a final point, if you are still struggling, it will help you if you make a quick graph of the two points and connect them with a line! From this you can tell if your gradient should be positive ( y increases as x moves to the right) or negative (y decreases as x moves to the right) and should also give you an idea of the steepness or magnitude of your gradient.