need help with a Math problem - linear equation
Hey guys,
so I'm learning geometry to help with game programming, I came across an anomaly but I'm wondering is it really an anomaly, I asked the question on reddit but nobody has got back to me with an answer, so I said I'll see if anybody on here can help :) so here is the question
I am following - https://www.khanacademy.org/math/geometry-home/cc-geometry-circles/arc-measures/v/arc-measure-with-equations-examples?modal=1 and attempting the first question in the video.
I figured out k, k = 5.25 - p/4
when I use the full equation of the whole circle I get
6k + 312 = 360 - 2p
=> 6(5.25 - p/4) + 312 = 360 - 2p
=> 31.25 - 1.5p + 312 = 360 - 2p
=> -1.p + 343.25 = 360 - 2p
=> 0.5p = 360 - 343.5
=> 0.5p = 16.5
=> p = 33
33 is indeed the right answer but I thought by using another equation we could also find P but it turns out they just prove they equal each other and I'm not finding P.
but when I try to use k in the top equation (4k + 159 = 180 - p)
they just prove they equal and fail to solve for P
4(5.25 - p/4) + 159 = 180 - p
=> 21 - p + 159 = 180 - p
=> -p + 180 = 180 - p
=> -p = 180 - 180 - p
=> -p + p = 0
=> 0 = 0
how come using one of these equations will solve for P and the other won't? is it something I am doing wrong in the latter example?
I also got p = 33 for 2k + 153 = 180 - p ( bottom equation ) but as mentioned the top equation just cancels out and proves they are equal to each other ( I fail to solve for p)
thanks
so I'm learning geometry to help with game programming, I came across an anomaly but I'm wondering is it really an anomaly, I asked the question on reddit but nobody has got back to me with an answer, so I said I'll see if anybody on here can help :) so here is the question
I am following - https://www.khanacademy.org/math/geometry-home/cc-geometry-circles/arc-measures/v/arc-measure-with-equations-examples?modal=1 and attempting the first question in the video.
I figured out k, k = 5.25 - p/4
when I use the full equation of the whole circle I get
6k + 312 = 360 - 2p
=> 6(5.25 - p/4) + 312 = 360 - 2p
=> 31.25 - 1.5p + 312 = 360 - 2p
=> -1.p + 343.25 = 360 - 2p
=> 0.5p = 360 - 343.5
=> 0.5p = 16.5
=> p = 33
33 is indeed the right answer but I thought by using another equation we could also find P but it turns out they just prove they equal each other and I'm not finding P.
but when I try to use k in the top equation (4k + 159 = 180 - p)
they just prove they equal and fail to solve for P
4(5.25 - p/4) + 159 = 180 - p
=> 21 - p + 159 = 180 - p
=> -p + 180 = 180 - p
=> -p = 180 - 180 - p
=> -p + p = 0
=> 0 = 0
how come using one of these equations will solve for P and the other won't? is it something I am doing wrong in the latter example?
I also got p = 33 for 2k + 153 = 180 - p ( bottom equation ) but as mentioned the top equation just cancels out and proves they are equal to each other ( I fail to solve for p)
thanks
parallel lines have no solution, intersecting lines have 1 solution, and the same line is the same line.
the second one is the same equation:
5.25*4 is 21. 21+159 is 180. 180-p = 180-p
there isn't anything more you can do with those 2 equations... they are really the same equation. You did it right, you just happen to have the same equation on both sides, and it isn't useful.
geometry has this problem frequently. Due to properties of lines and angles and all that good stuff, you can cheerfully show that 2 things are really the same thing and waste a lot of time. It isn't always obvious, but if you go back to the actual lines and study the properties of lines you can eventually prove that they really are the same thing. Skip that for now. Later, these same properties are pure gold... knowing that 2 things are the same and you know what one of them is, then you know what the other one is, and can show that …. doing no work to find your answer. So its as useful as it is annoying :)
the second one is the same equation:
5.25*4 is 21. 21+159 is 180. 180-p = 180-p
there isn't anything more you can do with those 2 equations... they are really the same equation. You did it right, you just happen to have the same equation on both sides, and it isn't useful.
geometry has this problem frequently. Due to properties of lines and angles and all that good stuff, you can cheerfully show that 2 things are really the same thing and waste a lot of time. It isn't always obvious, but if you go back to the actual lines and study the properties of lines you can eventually prove that they really are the same thing. Skip that for now. Later, these same properties are pure gold... knowing that 2 things are the same and you know what one of them is, then you know what the other one is, and can show that …. doing no work to find your answer. So its as useful as it is annoying :)
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In this case, graphically, you can see that in your first equation, the two equations have different slopes for p. The right-hand side has a slope of -3/2, and the left-hand side has a slope of -2. In other words, this means they intersect and don't run parallel forever.
In your latter example, the left-hand side has a slope of -1 (4 * -1/4) w.r.t. p, and the right-hand side also has a slope of -1. So both sides are the same line, just with a different offset, and won't intersect. So you can't solve it.
For example, if you have
3y = 2x [1]
6y = 4x [2]
When you solve a system of equations, each equation must be linearly independent.
You can clearly see that [2] is just a multiple of [1]. The equations are not linearly independent. If you were to try to "solve" this like you did above, you wouldn't be able to come to any conclusion. If you multiplied the top equation by 2 and then took the difference, you'd just get 0 = 0 again.
In your latter example, the left-hand side has a slope of -1 (4 * -1/4) w.r.t. p, and the right-hand side also has a slope of -1. So both sides are the same line, just with a different offset, and won't intersect. So you can't solve it.
For example, if you have
3y = 2x [1]
6y = 4x [2]
When you solve a system of equations, each equation must be linearly independent.
You can clearly see that [2] is just a multiple of [1]. The equations are not linearly independent. If you were to try to "solve" this like you did above, you wouldn't be able to come to any conclusion. If you multiplied the top equation by 2 and then took the difference, you'd just get 0 = 0 again.
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It looks like it's neither, in the example it looks like it's representing two degrees, which you equate (4k + 159)° = (2k + 153)°.
I actually don't know how the video linked is lining up with the with the numbers used in adam's post, so take what I wrote with a grain of salt...
my example was just using random equations, doesn't matter what the actual meaning was, I was just was trying to show the two equations don't give more information than just one equation, maybe it's not a good example.
I actually don't know how the video linked is lining up with the with the numbers used in adam's post, so take what I wrote with a grain of salt...
my example was just using random equations, doesn't matter what the actual meaning was, I was just was trying to show the two equations don't give more information than just one equation, maybe it's not a good example.
| I figured out k, k = 5.25 - p/4 |
What?
The only p in the problem is the "Circle P is below" in the first few frames.
I think you've got the wrong end of the stick, @adam2016.
Just solve 4k + 159 = 2k + 153 ... as they go to absolutely excruciating length to do!
thanks guys :)
yeah I don't think they are equations of a line, I think more we are just told to solve for k and find what the measure of the arc is, which would be the measure of the angle sub tending the arc in question
yeah I don't think they are equations of a line, I think more we are just told to solve for k and find what the measure of the arc is, which would be the measure of the angle sub tending the arc in question
Alas, I am not going to watch all those videos to figure out which one you are referring to (the one that auto-played for me was not it), and I am on my phone e, so...
Unfortunately, yours is a simple algebra problem. Look at what you are substituting, and where.
You already know k = 5.25 - p/4.
First substitution gets you p = 33.
Now just solve for k:
K = 5.25 - 33/4
Also, the equations you appear to be getting are arbitrary. They have nothing to do with the circles except that they are given to you as the measure of an angle.
The point is to recognize some properties of circles, such as all angles in a circle sum to 360 degrees, etc, in order to construct an system of equations that you can solve with only one or two variables.
(BTW, the two expressions you have presented here are, in fact, linear: y = mx + b, so you did recognize that correctly. Remember, that is not the only way to represent a line, though.)
Hope this helps.
Unfortunately, yours is a simple algebra problem. Look at what you are substituting, and where.
You already know k = 5.25 - p/4.
First substitution gets you p = 33.
Now just solve for k:
K = 5.25 - 33/4
Also, the equations you appear to be getting are arbitrary. They have nothing to do with the circles except that they are given to you as the measure of an angle.
The point is to recognize some properties of circles, such as all angles in a circle sum to 360 degrees, etc, in order to construct an system of equations that you can solve with only one or two variables.
(BTW, the two expressions you have presented here are, in fact, linear: y = mx + b, so you did recognize that correctly. Remember, that is not the only way to represent a line, though.)
Hope this helps.
often overlooked is that arc and angles are a % of a circle. If the question is how long the arc is, its a % of how long the full circle is... etc. A lot of this stuff is presented inefficiently to show how they figured it out in 7000BC, not the best way to actually find it... same will happen when you get to calc, they will show you a lot of useless stuff before they show you how to do anything.
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