y = mx + b
I know what each piece represents. What I am curious about is why multiplying the x coordinate by the slope then adding the y intercept always gets you the corresponding y coordinate.
I believe I have this, let me know if I understand it correctly please and thanks a bunch for your explanation.
So the first line says y = b + the change in y or "Slope" y
Took me a minute but that is saying the distance up to b (3), + the distance the slope y travels (4) gets you the y-coordinate (7).
Then you needed to prove that slope y was equal to mx
So you said that m is slope y over slope x.
Then I believe you replaced slope x with just x because their values were the same.
And this is the part where I got a little confused. But here goes:
You said that slope y over x (which is slope or in other words m) times x gets you slope y because the x's cross cancel.
Then you went on to say that slope y equals slope times slope x because the slope x's cross canceled and left you with slope y.
And since slope x is the same as x, you can replace slope x with x.
And all that was to prove that slope y = mx?
you have a good vision of how the equation is derived from the co-ordinate geometry.
I like to use the right-angled triangle as it is very useful for illustrations.
If instead, you use the term "delta", meaning a "small change" instead of "slope",
when referring to the perpendicular sides of the triangle,
you will be much clearer I'm sure.
The slope is the ratio of the vertical side to the horizontal side of the right-angled triangle.
Hence, it does not matter what size the triangle is.
The ratio of vertical side to horizontal side is constant.
This constant is called the slope of the line (like a hill with a slope going up or down).
Since "b" is on the y-axis, then in the drawing
Then, by rearranging the definition for the slope, using the drawing,
we get the line equation in "y=mx+b" form.
In the equation, the slope is "m"
That should help with understanding the equation derived from 2 points or a single point and the slope
Here's my essay-like attempt at answering your questions. It might not be what you want, but it might help too.
The best way to tackle a question like this is to look at what a number plane is, what a function is, and what things are like 'slope' and 'intercepts' are.
For an explicit function, the Xs and Ys can be thought of loosely like an 'input' and 'output'. You put a particular value for x into the equation, and it'll give you a value for y according to a rule. As if you put a number into a computer, and then it spits out another number which it manipulated according to some rule. In this case, y = mx + b is that 'rule'. A number plane can be seen as a graphical representation of this. It shows you that for a particular value of 'x', it'll give you a certain 'y'.
So with that out of the way, what is slope? Slope isn't exactly an abstract entity. A slope of 3 doesn't create the equation y = 3x. Rather, you need to think of it as the equation y = 3x creating the slope of 3. With something really simple such as y = x, whenever you put in a value for 'x' you will come out with the same value for 'y'. If x = 2, then y = 2. If you were to graph this on a number plane, you would get a diagonal line from bottom-left to top-right. The line has a slope of 1.
But with a function such as y = 4x, notice how our value for y changes as we change x. When x = 1, y = 4, and when x = 2, then y = 8. For every x we increase, the output y will increase by 4. Slope by definition is 'a change in y' over the corrisponding 'change in x'. So if y changes by 4 when we change x by 1:
What this demonstrates is that by multiplying the input x by some number, such as 4, it means that every time you increase that input by 1... it increases the output by what we're multiply x by. This multiplication of 'x' is what creates the slope of a line when you graph it.
Now, the y-intercept of a function is simply what y, that is the output, would equal when x = 0. Well for our function y = 4x, that's simply 0. Because 4 * 0 = 0. But what about y = x + 2? Well in that case, when x = 0, y will be 2. That's why the y-intercept is always equal to whatever constant is on the end of the equation. Because the mx will be 0, leaving only 'b' as your output for y.
Another way to look at it is to compare the functions y = 3x and y = 3x + 1. If we watched what happened for x = 0 to 5...
[y = 3x]
x = 0, y = 0
x = 1, y = 3
x = 2, y = 6
x = 3, y = 9
x = 4, y = 12
x = 5, y = 15
[y = 3x + 1]
x = 0, y = 1
x = 1, y = 4
x = 2, y = 7
x = 3, y = 10
x = 4, y = 13
x = 5, y = 16
When there is no constant on the end and you just have an equation y = mx, then the line will pass through (0, 0). If you add a constant on the end, such as + 1, then the entire line will be shifted up or down by that amount. In this case, the line and all its outputs are shifted up by 1. Because of that, where as the line previously cut the y-axis at (0, 0) - it now cuts at (0, 1), which is equal to b.
So in summary, the reason why y = mx + b works the way it does is because multiply 'x' by 'm' changes the rate at which y changes as you change x, giving you the slope on your line. And the reason 'b' is the y intercept is because when x = 0, the mx will equal zero as well, leaving y = b for the output.
Hope it helped. But if you already knew all this then that's fine too.