about equation of line.

crprogc

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Joined
Mar 24, 2011
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2
Hello,
This is my first post and I'll keep visit here to ask or answer if I know :)
I'm reading a book and I don't understand how this works.

x = Px + tdx
y = Py + tdy

and

\(\displaystyle y = dy\tfrac{x-Px}{dx} + Py\)

and it become a

\(\displaystyle 0 = \tfrac{y-Py}{dy} - \tfrac{x-Px}{dx}\)

and then it become a

= (-dy)x + (dx)y + (dyPx - dxPy)

= ax + by + c

I really don't understand is that how I can get

(-dy)x + (dx)y + (dyPx - dxPy)

from

\(\displaystyle 0 = \tfrac{y-Py}{dy} - \tfrac{x-Px}{dx}\)

I always get this

\(\displaystyle \frac{(y-Py)dx - (x-Px)dy}{dydx}\)

I don't know how to get rid of dydx in the bottom.

Thanks for help :)
 
1) Put the equation back. "0 ="
2) Realize that it is generally assumed dx is NOT zero and dy is NOT zero.
 
I am confused. You have posted under beginning algebra a question involving calculus. Have you studied calculus?
 
crprogc said:
x = Px + tdx [1]
y = Py + tdy [2]
Well, with no background given in your problem, then really simple:
[1] divide by x : P + td = 1
[2] divide by y : P + td = 1
:roll:
 
No, I didn't study calculus properly. I thought it is a algebra problem.
I think I should ask again differently.

There is a general form of equation of line which is

ax + by + c = 0

Do you guys know how to leading that general form?
Do I need to study calculus for this? or is there any web pages that explain this? (I searched but I couldn't find it)

Thanks
 
crprogc said:
No, I didn't study calculus properly. I thought it is a algebra problem.
I think I should ask again differently.

There is a general form of equation of line which is

ax + by + c = 0

Do you guys know how to leading that general form?
Do I need to study calculus for this? or is there any web pages that explain this? (I searched but I couldn't find it)

Thanks

I am still not sure I understand your question. If my answer is completely off topic or too simple or too hard, either reframe your question or say you want someone else to help you.

I gather that you are just beginning to study algebra and do not know calculus at all. Is that right?

Are you familiar with graphing, with drawing a picture of a relationship between numbers as a geometric figure? I assume that you are.

Your equation, ax + by + c = 0, is very general. It has two variables, x and y, and three constants, a, b, and c. Let's make it more concrete by picking some values for a, b, and c, say 2, -4, and 12. So a specific example of your general equation is 2x - 4y + 12 = 0. To graph this, we do what? We choose values of x or y and solve for the other variable. For example if x = 0, (2 * 0) - 4y + 12 = 0 simplifies to 4y = 12 or y = 3. So put a dot at the point in the plane corresponding to x = 0 and y = 3. If y = 0, 2x + [(- 4) * 0] + 12 = 0, or 2x = - 12, or x = - 6. So put a dot where y = 0 and x = - 6. Do this a few more times. You will find (unless you made a mistake) that you can put a ruler down and draw a straight line that covers all the dots.

Furthermore, had we chosen different constants and graphed an equation of the same form with those different constants, we would find that the dots associated with that equation also fell along a straight line. No matter how many examples we take, we find a straight line associated with this trpe of equation.

That is, each triplet of constants generates a straight line if the equation has this general form of ax + by + c = 0. Of course different triplets generate different straight lines, but they are are all straight lines. (Furthermore, equations that have a different form, say axy - b = 0, do not have graphs that are straight lines.)

Now what I have written is by no means a formal proof. If what you want is a formal proof that there is an isomorphism between a linear equation in two variables and a straight line in the Euclidean plane, I shall leave that to others to give. Has this been any help?
 
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