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For what value of $K$ are the points $A(1, -1)$, $B(K/2, K/3)$, and $C(4, 1)$ collinear? [closed]
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For what value of $K$ are the points $A(1, -1)$, $B(K/2, K/3)$, and $C(4, 1)$ collinear?
The equation of line $AC$ is $2x-3y=5$. But the point $B$ doesn't satisfy the equation.
algebra-precalculus analytic-geometry
edited Jul 21 at 17:21
AccidentalFourierTransform
1,077625
asked Jul 21 at 15:40
S.Nep
243
closed as off-topic by amWhy, Morgan Rodgers, Jyrki Lahtonen, Mostafa Ayaz, José Carlos Santos Jul 21 at 21:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РamWhy, Morgan Rodgers, Jyrki Lahtonen, Mostafa Ayaz, Jos̩ Carlos Santos
 |Â
show 16 more comments
up vote
2
down vote
favorite
For what value of $K$ are the points $A(1, -1)$, $B(K/2, K/3)$, and $C(4, 1)$ collinear?
The equation of line $AC$ is $2x-3y=5$. But the point $B$ doesn't satisfy the equation.
algebra-precalculus analytic-geometry
edited Jul 21 at 17:21
AccidentalFourierTransform
1,077625
asked Jul 21 at 15:40
S.Nep
243
closed as off-topic by amWhy, Morgan Rodgers, Jyrki Lahtonen, Mostafa Ayaz, José Carlos Santos Jul 21 at 21:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РamWhy, Morgan Rodgers, Jyrki Lahtonen, Mostafa Ayaz, Jos̩ Carlos Santos
2
Is there, perhaps, a unique value of $K$ for which $B$ does satisfy that equation?
– Jyrki Lahtonen
Jul 21 at 15:43
3
When you are done thinking, please study our guide to new askers. Following those guidelines will do wonders in avoiding negative attention!
– Jyrki Lahtonen
Jul 21 at 15:45
1
Looks like this may be a trick question :-)
– Jyrki Lahtonen
Jul 21 at 15:48
2
Yes. I too think so
– S.Nep
Jul 21 at 15:57
2
@fleablood Of course "there is none" is a valid answer - so from that point of view, the question is not a trick question. But do you have an example of a trick question then? If someone asks me, when did I stop stealing candy from the children on my street? The answer is never.
– peter a g
Jul 21 at 16:14
 |Â
show 16 more comments
up vote
2
down vote
favorite
up vote
2
down vote
favorite
For what value of $K$ are the points $A(1, -1)$, $B(K/2, K/3)$, and $C(4, 1)$ collinear?
The equation of line $AC$ is $2x-3y=5$. But the point $B$ doesn't satisfy the equation.
algebra-precalculus analytic-geometry
edited Jul 21 at 17:21
AccidentalFourierTransform
1,077625
asked Jul 21 at 15:40
S.Nep
243
For what value of $K$ are the points $A(1, -1)$, $B(K/2, K/3)$, and $C(4, 1)$ collinear?
The equation of line $AC$ is $2x-3y=5$. But the point $B$ doesn't satisfy the equation.
algebra-precalculus analytic-geometry
edited Jul 21 at 17:21
AccidentalFourierTransform
1,077625
asked Jul 21 at 15:40
S.Nep
243
edited Jul 21 at 17:21
AccidentalFourierTransform
1,077625
edited Jul 21 at 17:21
AccidentalFourierTransform
1,077625
edited Jul 21 at 17:21
AccidentalFourierTransform
1,077625
1,077625
asked Jul 21 at 15:40
S.Nep
243
asked Jul 21 at 15:40
S.Nep
243
asked Jul 21 at 15:40
S.Nep
243
243
closed as off-topic by amWhy, Morgan Rodgers, Jyrki Lahtonen, Mostafa Ayaz, José Carlos Santos Jul 21 at 21:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РamWhy, Morgan Rodgers, Jyrki Lahtonen, Mostafa Ayaz, Jos̩ Carlos Santos
closed as off-topic by amWhy, Morgan Rodgers, Jyrki Lahtonen, Mostafa Ayaz, José Carlos Santos Jul 21 at 21:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РamWhy, Morgan Rodgers, Jyrki Lahtonen, Mostafa Ayaz, Jos̩ Carlos Santos
2
Is there, perhaps, a unique value of $K$ for which $B$ does satisfy that equation?
– Jyrki Lahtonen
Jul 21 at 15:43
3
When you are done thinking, please study our guide to new askers. Following those guidelines will do wonders in avoiding negative attention!
– Jyrki Lahtonen
Jul 21 at 15:45
1
Looks like this may be a trick question :-)
– Jyrki Lahtonen
Jul 21 at 15:48
2
Yes. I too think so
– S.Nep
Jul 21 at 15:57
2
@fleablood Of course "there is none" is a valid answer - so from that point of view, the question is not a trick question. But do you have an example of a trick question then? If someone asks me, when did I stop stealing candy from the children on my street? The answer is never.
– peter a g
Jul 21 at 16:14
 |Â
show 16 more comments
2
Is there, perhaps, a unique value of $K$ for which $B$ does satisfy that equation?
– Jyrki Lahtonen
Jul 21 at 15:43
3
When you are done thinking, please study our guide to new askers. Following those guidelines will do wonders in avoiding negative attention!
– Jyrki Lahtonen
Jul 21 at 15:45
1
Looks like this may be a trick question :-)
– Jyrki Lahtonen
Jul 21 at 15:48
2
Yes. I too think so
– S.Nep
Jul 21 at 15:57
2
@fleablood Of course "there is none" is a valid answer - so from that point of view, the question is not a trick question. But do you have an example of a trick question then? If someone asks me, when did I stop stealing candy from the children on my street? The answer is never.
– peter a g
Jul 21 at 16:14
2
2
Is there, perhaps, a unique value of $K$ for which $B$ does satisfy that equation?
– Jyrki Lahtonen
Jul 21 at 15:43
Is there, perhaps, a unique value of $K$ for which $B$ does satisfy that equation?
– Jyrki Lahtonen
Jul 21 at 15:43
3
3
When you are done thinking, please study our guide to new askers. Following those guidelines will do wonders in avoiding negative attention!
– Jyrki Lahtonen
Jul 21 at 15:45
When you are done thinking, please study our guide to new askers. Following those guidelines will do wonders in avoiding negative attention!
– Jyrki Lahtonen
Jul 21 at 15:45
1
1
Looks like this may be a trick question :-)
– Jyrki Lahtonen
Jul 21 at 15:48
Looks like this may be a trick question :-)
– Jyrki Lahtonen
Jul 21 at 15:48
2
2
Yes. I too think so
– S.Nep
Jul 21 at 15:57
Yes. I too think so
– S.Nep
Jul 21 at 15:57
2
2
@fleablood Of course "there is none" is a valid answer - so from that point of view, the question is not a trick question. But do you have an example of a trick question then? If someone asks me, when did I stop stealing candy from the children on my street? The answer is never.
– peter a g
Jul 21 at 16:14
@fleablood Of course "there is none" is a valid answer - so from that point of view, the question is not a trick question. But do you have an example of a trick question then? If someone asks me, when did I stop stealing candy from the children on my street? The answer is never.
– peter a g
Jul 21 at 16:14
 |Â
show 16 more comments
3 Answers
3
active
oldest
votes
up vote
7
down vote
"The equation of line AC is 2x-3y=5. But the point B doesnt satisfy the equation."
WHy not?
If that is the equation of the line and if we are told that $(frac K2, frac K3)$ is a point of the line, then it must be true that $2*frac K2 - 3*frac K3 = 5$. For what value of $K$ is that true?
1) Is that the equation of the line?
The slope of the equation of the line is $m = frac C_y - A_yC_x - A_x = frac 1-(-1)4-1 = frac 23$ so the equation is $(y-A_y) = m*(x-A_x)$ so $y-(-1) = frac 23(x-1)$ or $y = frac 23 x -frac 53$ or as you put it $2x - 3y = 5$.
2) What value of $K$ satisfies $2*frac K2 - 3*frac K3 = 5$ ?
$K - K = 5$
$0 = 5$
There is no value of $K$ that satisfies.
So the answer to "For what value of K are the points A(1, -1), B(K/2, K/3), and C(4, 1) collinear?" is...
"None".
Which is a perfectly acceptable and valid answer. Just because a textbook asks a question doesn't mean that there is an valid answer. But in this case "none" is a valid answer. Just because a textbook asks "when does this occur" doesn't mean the answer isn't "never".
answered Jul 21 at 16:06
fleablood
60.4k22575
1
I don't think "none" is a valid answer to a question of the form "For what value of K ... ?". It would be a valid answer if "value" was plural: "For what values of K ... ?". Using the singular form implies that there is exactly one value; and is not a valid question if there are no values or more than one value.
– Paulpro
Jul 21 at 16:54
Generally speaking I agree with your comments on none being a valid answer. For which I upvoted. However I want to add that the textbook is partially at fault here as well. "For what value P" carries an implication that there is a unique value satisfying P, whereas a phrasing like "For what values P" carries no such implications (to my ears anyway). So while I certainly agree that "there is no such value," is unambiguously the correct answer to the question the textbook poses, I think it would be fair to note that the textbook's phrasing suggests that answer would have to be wrong.
– jgon
Jul 21 at 16:57
Mmmm.... In general "What value K where whe $(aK, cK)$ is on the line $y = mx + b$" is solvable by $cK= maK + b$ so $K = frac bc-am$ will have one solutions. When $c=am$ are special exceptions where if $b =0$ there are infinite solutions an $bne 0$ there are none, are cases the student should be expected to be aware of and able to respond to. So I accept it. I suppose it's valid to say it is misleading. But it's not anything that should run up the psychology bills 20 years later.
– fleablood
Jul 21 at 19:26
add a comment |Â
up vote
0
down vote
If can you can proof that no real number $K$ will lead to a point $B$ which is on the line through $A$ and $C$, that is fine.
It might be an error in the book or for educational purposes.
My attempt:
The line through $A$ and $C$ is
$$
(1-lambda) A + lambda C quad (lambda in mathbbR)
$$
We equate with point $B$ and get
$$
(1 - lambda) (1, -1) + lambda (4,1) = (K/2, K/3)
$$
which gives the system
$$
(1 - lambda) + 4 lambda = K/2 \
-(1-lambda) + lambda = K/3
$$
which is equivalent to the inhomogeneous system in unknowns $lambda$ and $K$:
$$
left[
beginarrayr
3 & -1/2 & -1 \
2 & -1/3 & 1
endarray
right]
to
left[
beginarrayr
1 & -1/6 & -2 \
2 & -1/3 & 1
endarray
right]
to
left[
beginarrayr
1 & -1/6 & -2 \
0 & 0 & 5
endarray
right]
$$
We subtract the second row from the first row, then subtract two times the first row from the second row. The resulting last row is inconsistent ($0cdot lambda + 0 cdot K = 5$), so there is no solution.
answered Jul 21 at 16:14
mvw
30.4k22250
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up vote
0
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The direction vector of the straight line $K(1/2,1/3)$ is obviously parallel to the direction vector $C-A=(3,2)$ of the straight line through $A$ and $C$.
answered Jul 21 at 16:44
Michael Hoppe
9,55631432
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
"The equation of line AC is 2x-3y=5. But the point B doesnt satisfy the equation."
WHy not?
If that is the equation of the line and if we are told that $(frac K2, frac K3)$ is a point of the line, then it must be true that $2*frac K2 - 3*frac K3 = 5$. For what value of $K$ is that true?
1) Is that the equation of the line?
The slope of the equation of the line is $m = frac C_y - A_yC_x - A_x = frac 1-(-1)4-1 = frac 23$ so the equation is $(y-A_y) = m*(x-A_x)$ so $y-(-1) = frac 23(x-1)$ or $y = frac 23 x -frac 53$ or as you put it $2x - 3y = 5$.
2) What value of $K$ satisfies $2*frac K2 - 3*frac K3 = 5$ ?
$K - K = 5$
$0 = 5$
There is no value of $K$ that satisfies.
So the answer to "For what value of K are the points A(1, -1), B(K/2, K/3), and C(4, 1) collinear?" is...
"None".
Which is a perfectly acceptable and valid answer. Just because a textbook asks a question doesn't mean that there is an valid answer. But in this case "none" is a valid answer. Just because a textbook asks "when does this occur" doesn't mean the answer isn't "never".
answered Jul 21 at 16:06
fleablood
60.4k22575
1
I don't think "none" is a valid answer to a question of the form "For what value of K ... ?". It would be a valid answer if "value" was plural: "For what values of K ... ?". Using the singular form implies that there is exactly one value; and is not a valid question if there are no values or more than one value.
– Paulpro
Jul 21 at 16:54
Generally speaking I agree with your comments on none being a valid answer. For which I upvoted. However I want to add that the textbook is partially at fault here as well. "For what value P" carries an implication that there is a unique value satisfying P, whereas a phrasing like "For what values P" carries no such implications (to my ears anyway). So while I certainly agree that "there is no such value," is unambiguously the correct answer to the question the textbook poses, I think it would be fair to note that the textbook's phrasing suggests that answer would have to be wrong.
– jgon
Jul 21 at 16:57
Mmmm.... In general "What value K where whe $(aK, cK)$ is on the line $y = mx + b$" is solvable by $cK= maK + b$ so $K = frac bc-am$ will have one solutions. When $c=am$ are special exceptions where if $b =0$ there are infinite solutions an $bne 0$ there are none, are cases the student should be expected to be aware of and able to respond to. So I accept it. I suppose it's valid to say it is misleading. But it's not anything that should run up the psychology bills 20 years later.
– fleablood
Jul 21 at 19:26
add a comment |Â
up vote
7
down vote
"The equation of line AC is 2x-3y=5. But the point B doesnt satisfy the equation."
WHy not?
If that is the equation of the line and if we are told that $(frac K2, frac K3)$ is a point of the line, then it must be true that $2*frac K2 - 3*frac K3 = 5$. For what value of $K$ is that true?
1) Is that the equation of the line?
The slope of the equation of the line is $m = frac C_y - A_yC_x - A_x = frac 1-(-1)4-1 = frac 23$ so the equation is $(y-A_y) = m*(x-A_x)$ so $y-(-1) = frac 23(x-1)$ or $y = frac 23 x -frac 53$ or as you put it $2x - 3y = 5$.
2) What value of $K$ satisfies $2*frac K2 - 3*frac K3 = 5$ ?
$K - K = 5$
$0 = 5$
There is no value of $K$ that satisfies.
So the answer to "For what value of K are the points A(1, -1), B(K/2, K/3), and C(4, 1) collinear?" is...
"None".
Which is a perfectly acceptable and valid answer. Just because a textbook asks a question doesn't mean that there is an valid answer. But in this case "none" is a valid answer. Just because a textbook asks "when does this occur" doesn't mean the answer isn't "never".
answered Jul 21 at 16:06
fleablood
60.4k22575
1
I don't think "none" is a valid answer to a question of the form "For what value of K ... ?". It would be a valid answer if "value" was plural: "For what values of K ... ?". Using the singular form implies that there is exactly one value; and is not a valid question if there are no values or more than one value.
– Paulpro
Jul 21 at 16:54
Generally speaking I agree with your comments on none being a valid answer. For which I upvoted. However I want to add that the textbook is partially at fault here as well. "For what value P" carries an implication that there is a unique value satisfying P, whereas a phrasing like "For what values P" carries no such implications (to my ears anyway). So while I certainly agree that "there is no such value," is unambiguously the correct answer to the question the textbook poses, I think it would be fair to note that the textbook's phrasing suggests that answer would have to be wrong.
– jgon
Jul 21 at 16:57
Mmmm.... In general "What value K where whe $(aK, cK)$ is on the line $y = mx + b$" is solvable by $cK= maK + b$ so $K = frac bc-am$ will have one solutions. When $c=am$ are special exceptions where if $b =0$ there are infinite solutions an $bne 0$ there are none, are cases the student should be expected to be aware of and able to respond to. So I accept it. I suppose it's valid to say it is misleading. But it's not anything that should run up the psychology bills 20 years later.
– fleablood
Jul 21 at 19:26
add a comment |Â
up vote
7
down vote
up vote
7
down vote
"The equation of line AC is 2x-3y=5. But the point B doesnt satisfy the equation."
WHy not?
If that is the equation of the line and if we are told that $(frac K2, frac K3)$ is a point of the line, then it must be true that $2*frac K2 - 3*frac K3 = 5$. For what value of $K$ is that true?
1) Is that the equation of the line?
The slope of the equation of the line is $m = frac C_y - A_yC_x - A_x = frac 1-(-1)4-1 = frac 23$ so the equation is $(y-A_y) = m*(x-A_x)$ so $y-(-1) = frac 23(x-1)$ or $y = frac 23 x -frac 53$ or as you put it $2x - 3y = 5$.
2) What value of $K$ satisfies $2*frac K2 - 3*frac K3 = 5$ ?
$K - K = 5$
$0 = 5$
There is no value of $K$ that satisfies.
So the answer to "For what value of K are the points A(1, -1), B(K/2, K/3), and C(4, 1) collinear?" is...
"None".
Which is a perfectly acceptable and valid answer. Just because a textbook asks a question doesn't mean that there is an valid answer. But in this case "none" is a valid answer. Just because a textbook asks "when does this occur" doesn't mean the answer isn't "never".
answered Jul 21 at 16:06
fleablood
60.4k22575
"The equation of line AC is 2x-3y=5. But the point B doesnt satisfy the equation."
WHy not?
If that is the equation of the line and if we are told that $(frac K2, frac K3)$ is a point of the line, then it must be true that $2*frac K2 - 3*frac K3 = 5$. For what value of $K$ is that true?
1) Is that the equation of the line?
The slope of the equation of the line is $m = frac C_y - A_yC_x - A_x = frac 1-(-1)4-1 = frac 23$ so the equation is $(y-A_y) = m*(x-A_x)$ so $y-(-1) = frac 23(x-1)$ or $y = frac 23 x -frac 53$ or as you put it $2x - 3y = 5$.
2) What value of $K$ satisfies $2*frac K2 - 3*frac K3 = 5$ ?
$K - K = 5$
$0 = 5$
There is no value of $K$ that satisfies.
So the answer to "For what value of K are the points A(1, -1), B(K/2, K/3), and C(4, 1) collinear?" is...
"None".
Which is a perfectly acceptable and valid answer. Just because a textbook asks a question doesn't mean that there is an valid answer. But in this case "none" is a valid answer. Just because a textbook asks "when does this occur" doesn't mean the answer isn't "never".
answered Jul 21 at 16:06
fleablood
60.4k22575
answered Jul 21 at 16:06
fleablood
60.4k22575
answered Jul 21 at 16:06
fleablood
60.4k22575
answered Jul 21 at 16:06
fleablood
60.4k22575
60.4k22575
1
I don't think "none" is a valid answer to a question of the form "For what value of K ... ?". It would be a valid answer if "value" was plural: "For what values of K ... ?". Using the singular form implies that there is exactly one value; and is not a valid question if there are no values or more than one value.
– Paulpro
Jul 21 at 16:54
Generally speaking I agree with your comments on none being a valid answer. For which I upvoted. However I want to add that the textbook is partially at fault here as well. "For what value P" carries an implication that there is a unique value satisfying P, whereas a phrasing like "For what values P" carries no such implications (to my ears anyway). So while I certainly agree that "there is no such value," is unambiguously the correct answer to the question the textbook poses, I think it would be fair to note that the textbook's phrasing suggests that answer would have to be wrong.
– jgon
Jul 21 at 16:57
Mmmm.... In general "What value K where whe $(aK, cK)$ is on the line $y = mx + b$" is solvable by $cK= maK + b$ so $K = frac bc-am$ will have one solutions. When $c=am$ are special exceptions where if $b =0$ there are infinite solutions an $bne 0$ there are none, are cases the student should be expected to be aware of and able to respond to. So I accept it. I suppose it's valid to say it is misleading. But it's not anything that should run up the psychology bills 20 years later.
– fleablood
Jul 21 at 19:26
add a comment |Â
1
I don't think "none" is a valid answer to a question of the form "For what value of K ... ?". It would be a valid answer if "value" was plural: "For what values of K ... ?". Using the singular form implies that there is exactly one value; and is not a valid question if there are no values or more than one value.
– Paulpro
Jul 21 at 16:54
Generally speaking I agree with your comments on none being a valid answer. For which I upvoted. However I want to add that the textbook is partially at fault here as well. "For what value P" carries an implication that there is a unique value satisfying P, whereas a phrasing like "For what values P" carries no such implications (to my ears anyway). So while I certainly agree that "there is no such value," is unambiguously the correct answer to the question the textbook poses, I think it would be fair to note that the textbook's phrasing suggests that answer would have to be wrong.
– jgon
Jul 21 at 16:57
Mmmm.... In general "What value K where whe $(aK, cK)$ is on the line $y = mx + b$" is solvable by $cK= maK + b$ so $K = frac bc-am$ will have one solutions. When $c=am$ are special exceptions where if $b =0$ there are infinite solutions an $bne 0$ there are none, are cases the student should be expected to be aware of and able to respond to. So I accept it. I suppose it's valid to say it is misleading. But it's not anything that should run up the psychology bills 20 years later.
– fleablood
Jul 21 at 19:26
1
1
I don't think "none" is a valid answer to a question of the form "For what value of K ... ?". It would be a valid answer if "value" was plural: "For what values of K ... ?". Using the singular form implies that there is exactly one value; and is not a valid question if there are no values or more than one value.
– Paulpro
Jul 21 at 16:54
I don't think "none" is a valid answer to a question of the form "For what value of K ... ?". It would be a valid answer if "value" was plural: "For what values of K ... ?". Using the singular form implies that there is exactly one value; and is not a valid question if there are no values or more than one value.
– Paulpro
Jul 21 at 16:54
Generally speaking I agree with your comments on none being a valid answer. For which I upvoted. However I want to add that the textbook is partially at fault here as well. "For what value P" carries an implication that there is a unique value satisfying P, whereas a phrasing like "For what values P" carries no such implications (to my ears anyway). So while I certainly agree that "there is no such value," is unambiguously the correct answer to the question the textbook poses, I think it would be fair to note that the textbook's phrasing suggests that answer would have to be wrong.
– jgon
Jul 21 at 16:57
Generally speaking I agree with your comments on none being a valid answer. For which I upvoted. However I want to add that the textbook is partially at fault here as well. "For what value P" carries an implication that there is a unique value satisfying P, whereas a phrasing like "For what values P" carries no such implications (to my ears anyway). So while I certainly agree that "there is no such value," is unambiguously the correct answer to the question the textbook poses, I think it would be fair to note that the textbook's phrasing suggests that answer would have to be wrong.
– jgon
Jul 21 at 16:57
Mmmm.... In general "What value K where whe $(aK, cK)$ is on the line $y = mx + b$" is solvable by $cK= maK + b$ so $K = frac bc-am$ will have one solutions. When $c=am$ are special exceptions where if $b =0$ there are infinite solutions an $bne 0$ there are none, are cases the student should be expected to be aware of and able to respond to. So I accept it. I suppose it's valid to say it is misleading. But it's not anything that should run up the psychology bills 20 years later.
– fleablood
Jul 21 at 19:26
Mmmm.... In general "What value K where whe $(aK, cK)$ is on the line $y = mx + b$" is solvable by $cK= maK + b$ so $K = frac bc-am$ will have one solutions. When $c=am$ are special exceptions where if $b =0$ there are infinite solutions an $bne 0$ there are none, are cases the student should be expected to be aware of and able to respond to. So I accept it. I suppose it's valid to say it is misleading. But it's not anything that should run up the psychology bills 20 years later.
– fleablood
Jul 21 at 19:26
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If can you can proof that no real number $K$ will lead to a point $B$ which is on the line through $A$ and $C$, that is fine.
It might be an error in the book or for educational purposes.
My attempt:
The line through $A$ and $C$ is
$$
(1-lambda) A + lambda C quad (lambda in mathbbR)
$$
We equate with point $B$ and get
$$
(1 - lambda) (1, -1) + lambda (4,1) = (K/2, K/3)
$$
which gives the system
$$
(1 - lambda) + 4 lambda = K/2 \
-(1-lambda) + lambda = K/3
$$
which is equivalent to the inhomogeneous system in unknowns $lambda$ and $K$:
$$
left[
beginarrayr
3 & -1/2 & -1 \
2 & -1/3 & 1
endarray
right]
to
left[
beginarrayr
1 & -1/6 & -2 \
2 & -1/3 & 1
endarray
right]
to
left[
beginarrayr
1 & -1/6 & -2 \
0 & 0 & 5
endarray
right]
$$
We subtract the second row from the first row, then subtract two times the first row from the second row. The resulting last row is inconsistent ($0cdot lambda + 0 cdot K = 5$), so there is no solution.
answered Jul 21 at 16:14
mvw
30.4k22250
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up vote
0
down vote
If can you can proof that no real number $K$ will lead to a point $B$ which is on the line through $A$ and $C$, that is fine.
It might be an error in the book or for educational purposes.
My attempt:
The line through $A$ and $C$ is
$$
(1-lambda) A + lambda C quad (lambda in mathbbR)
$$
We equate with point $B$ and get
$$
(1 - lambda) (1, -1) + lambda (4,1) = (K/2, K/3)
$$
which gives the system
$$
(1 - lambda) + 4 lambda = K/2 \
-(1-lambda) + lambda = K/3
$$
which is equivalent to the inhomogeneous system in unknowns $lambda$ and $K$:
$$
left[
beginarrayr
3 & -1/2 & -1 \
2 & -1/3 & 1
endarray
right]
to
left[
beginarrayr
1 & -1/6 & -2 \
2 & -1/3 & 1
endarray
right]
to
left[
beginarrayr
1 & -1/6 & -2 \
0 & 0 & 5
endarray
right]
$$
We subtract the second row from the first row, then subtract two times the first row from the second row. The resulting last row is inconsistent ($0cdot lambda + 0 cdot K = 5$), so there is no solution.
answered Jul 21 at 16:14
mvw
30.4k22250
add a comment |Â
up vote
0
down vote
up vote
0
down vote
If can you can proof that no real number $K$ will lead to a point $B$ which is on the line through $A$ and $C$, that is fine.
It might be an error in the book or for educational purposes.
My attempt:
The line through $A$ and $C$ is
$$
(1-lambda) A + lambda C quad (lambda in mathbbR)
$$
We equate with point $B$ and get
$$
(1 - lambda) (1, -1) + lambda (4,1) = (K/2, K/3)
$$
which gives the system
$$
(1 - lambda) + 4 lambda = K/2 \
-(1-lambda) + lambda = K/3
$$
which is equivalent to the inhomogeneous system in unknowns $lambda$ and $K$:
$$
left[
beginarrayr
3 & -1/2 & -1 \
2 & -1/3 & 1
endarray
right]
to
left[
beginarrayr
1 & -1/6 & -2 \
2 & -1/3 & 1
endarray
right]
to
left[
beginarrayr
1 & -1/6 & -2 \
0 & 0 & 5
endarray
right]
$$
We subtract the second row from the first row, then subtract two times the first row from the second row. The resulting last row is inconsistent ($0cdot lambda + 0 cdot K = 5$), so there is no solution.
answered Jul 21 at 16:14
mvw
30.4k22250
If can you can proof that no real number $K$ will lead to a point $B$ which is on the line through $A$ and $C$, that is fine.
It might be an error in the book or for educational purposes.
My attempt:
The line through $A$ and $C$ is
$$
(1-lambda) A + lambda C quad (lambda in mathbbR)
$$
We equate with point $B$ and get
$$
(1 - lambda) (1, -1) + lambda (4,1) = (K/2, K/3)
$$
which gives the system
$$
(1 - lambda) + 4 lambda = K/2 \
-(1-lambda) + lambda = K/3
$$
which is equivalent to the inhomogeneous system in unknowns $lambda$ and $K$:
$$
left[
beginarrayr
3 & -1/2 & -1 \
2 & -1/3 & 1
endarray
right]
to
left[
beginarrayr
1 & -1/6 & -2 \
2 & -1/3 & 1
endarray
right]
to
left[
beginarrayr
1 & -1/6 & -2 \
0 & 0 & 5
endarray
right]
$$
We subtract the second row from the first row, then subtract two times the first row from the second row. The resulting last row is inconsistent ($0cdot lambda + 0 cdot K = 5$), so there is no solution.
answered Jul 21 at 16:14
mvw
30.4k22250
edited Jul 21 at 16:33
answered Jul 21 at 16:14
mvw
30.4k22250
answered Jul 21 at 16:14
mvw
30.4k22250
answered Jul 21 at 16:14
mvw
30.4k22250
30.4k22250
add a comment |Â
add a comment |Â
up vote
0
down vote
The direction vector of the straight line $K(1/2,1/3)$ is obviously parallel to the direction vector $C-A=(3,2)$ of the straight line through $A$ and $C$.
answered Jul 21 at 16:44
Michael Hoppe
9,55631432
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0
down vote
The direction vector of the straight line $K(1/2,1/3)$ is obviously parallel to the direction vector $C-A=(3,2)$ of the straight line through $A$ and $C$.
answered Jul 21 at 16:44
Michael Hoppe
9,55631432
add a comment |Â
up vote
0
down vote
up vote
0
down vote
The direction vector of the straight line $K(1/2,1/3)$ is obviously parallel to the direction vector $C-A=(3,2)$ of the straight line through $A$ and $C$.
answered Jul 21 at 16:44
Michael Hoppe
9,55631432
The direction vector of the straight line $K(1/2,1/3)$ is obviously parallel to the direction vector $C-A=(3,2)$ of the straight line through $A$ and $C$.
answered Jul 21 at 16:44
Michael Hoppe
9,55631432
edited Jul 21 at 16:55
answered Jul 21 at 16:44
Michael Hoppe
9,55631432
answered Jul 21 at 16:44
Michael Hoppe
9,55631432
answered Jul 21 at 16:44
Michael Hoppe
9,55631432
9,55631432
add a comment |Â
add a comment |Â
2
Is there, perhaps, a unique value of $K$ for which $B$ does satisfy that equation?
– Jyrki Lahtonen
Jul 21 at 15:43
3
When you are done thinking, please study our guide to new askers. Following those guidelines will do wonders in avoiding negative attention!
– Jyrki Lahtonen
Jul 21 at 15:45
1
Looks like this may be a trick question :-)
– Jyrki Lahtonen
Jul 21 at 15:48
2
Yes. I too think so
– S.Nep
Jul 21 at 15:57
2
@fleablood Of course "there is none" is a valid answer - so from that point of view, the question is not a trick question. But do you have an example of a trick question then? If someone asks me, when did I stop stealing candy from the children on my street? The answer is never.
– peter a g
Jul 21 at 16:14