4 Line equations - Really stuck - pls guide me
- Thread starter Gretl
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I kind of wonder if the poor terminology used here is part of why you're confused. That problem statement seems like it's quite terribly written. "These four lines meet in 6 places" could be reasonably interpreted as saying that all four of the lines pass through six points. But, given that you say you've solved part (a), so I'm sure in your working you found that's not the case - it actually means that there are six points where two of the six lines intersect. Part (b) asks you to go the other way around and find four lines such that two of them intersect at each of these points.
As for how to begin the problem, I'd start by plotting the six given points on a graph. Immediately, I can see that there are three "nice" points that look like they're probably on the same line. Which three points are those? Taking any two of those points, what is the slope of the line passing through both of them? What is the y-intercept of that line? Putting those two together, what is the line passing through them? Now, I also see two other "nice" points that look like they probably lie on a line together. Which two points do you think those are? What's the slope of the line passing through them? What's the y-intercept? This gives you two lines. Do these two lines intersect? If so, do they intersect at one of the six given points? What do you think your next step should be?
If you get stuck again, that's okay, but when you reply please include any and all work you've done on this problem, even if you know it's wrong. Thank you.
As for how to begin the problem, I'd start by plotting the six given points on a graph. Immediately, I can see that there are three "nice" points that look like they're probably on the same line. Which three points are those? Taking any two of those points, what is the slope of the line passing through both of them? What is the y-intercept of that line? Putting those two together, what is the line passing through them? Now, I also see two other "nice" points that look like they probably lie on a line together. Which two points do you think those are? What's the slope of the line passing through them? What's the y-intercept? This gives you two lines. Do these two lines intersect? If so, do they intersect at one of the six given points? What do you think your next step should be?
If you get stuck again, that's okay, but when you reply please include any and all work you've done on this problem, even if you know it's wrong. Thank you.
Otis
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It looks like you took the picture before finishing part (a). If later you drew the fourth line, and extended all lines as needed, then you probably saw a total of six intersection points.
To start part (b), plot the points. Then experiment! Try to form four lines such that you end up with a total of six intersection points. If your first attempt fails, then start over on a new sheet and try again.
Once you find four lines and six intersection points, you can determine the four equations using any method that you've learned thus far for writing the equation of a line knowing the coordinates of two points.
To start part (b), plot the points. Then experiment! Try to form four lines such that you end up with a total of six intersection points. If your first attempt fails, then start over on a new sheet and try again.
Once you find four lines and six intersection points, you can determine the four equations using any method that you've learned thus far for writing the equation of a line knowing the coordinates of two points.
Steven G
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I think that you need to understand that to have the 4 lines meet in 6 places that each line need to cross the other three line.
If we call the lines L1, L2, L3 and L4 and let LiLj denote that Li intersects Lj
then if L1L2 L1L3 L1L4 L2L3 L2 L4 and L3L4 we have 6 points of intersection.
Personally I strongly dislike this problem and do not see much value in it.
If we call the lines L1, L2, L3 and L4 and let LiLj denote that Li intersects Lj
then if L1L2 L1L3 L1L4 L2L3 L2 L4 and L3L4 we have 6 points of intersection.
Personally I strongly dislike this problem and do not see much value in it.
Before we dive into the problem, I must say there are some interesting problem solving aspects that arise from trying to solve this problem. Especially if, like me, you try to generalize every problem you solve, then one quickly descends into some moderately tricky inductive proofs. But alas, none of this is relevant to solving this problem, so I'll leave it at that.
You already solved the first part of the exercise on your own. Nice job, by the way :-D. I've also implemented the graphs you drew in a graphing environment called Desmos; the URL to that is https://www.desmos.com/calculator/abaonibhgh. It includes notes on how to solve this problem, albeit that they are quite terse; I wrote it more elaborately at first, but I lost the results due to refreshing when I had no internet connection.
Now to the open part of this problem. To solve it, I have again implemented the exercise in that same graphing environment, this time with more elaborate notes on how to solve the problem. The link to that is https://www.desmos.com/calculator/gfxb4pkm3j. I've also purposefully left some parts open to the reader. In summary, you should look for triplets of intersection points that are collinear with one another, which simply means the points all lie on the same line. Note that I give no guarantee that this always leads to the right results; I merely argue it might. And the way I did it, it does.
The key take-away here is that finding lines based on certain points that meet certain conditions is done by analyzing the collinearity between certain sets of points. How exactly one should do that is, of course, dependent on the problem.
If you are still stuck or have proceeded to solve the exercise, please let us know.
Good luck!
You already solved the first part of the exercise on your own. Nice job, by the way :-D. I've also implemented the graphs you drew in a graphing environment called Desmos; the URL to that is https://www.desmos.com/calculator/abaonibhgh. It includes notes on how to solve this problem, albeit that they are quite terse; I wrote it more elaborately at first, but I lost the results due to refreshing when I had no internet connection.
Now to the open part of this problem. To solve it, I have again implemented the exercise in that same graphing environment, this time with more elaborate notes on how to solve the problem. The link to that is https://www.desmos.com/calculator/gfxb4pkm3j. I've also purposefully left some parts open to the reader. In summary, you should look for triplets of intersection points that are collinear with one another, which simply means the points all lie on the same line. Note that I give no guarantee that this always leads to the right results; I merely argue it might. And the way I did it, it does.
The key take-away here is that finding lines based on certain points that meet certain conditions is done by analyzing the collinearity between certain sets of points. How exactly one should do that is, of course, dependent on the problem.
If you are still stuck or have proceeded to solve the exercise, please let us know.
Good luck!
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