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In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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In the xy-coordinate plane, which of the following points must lie on the line kx + 3y = 6 for every possible value of k?

(A) (1,1)
(B) (0,2)
(C) (2,0)
(D) (3,6)
(E) (6,3)

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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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Bunuel wrote:
In the xy-coordinate plane, which of the following points must lie on the line kx + 3y = 6 for every possible value of k?

(A) (1,1)
(B) (0,2)
(C) (2,0)
(D) (3,6)
(E) (6,3)

Kudos for a correct solution.

We're looking for a set of coordinates that are not affected by the value of k.
Notice that, in answer choice B (0,2), the x-coordinate is 0 and y = 2
So, if we plug x = 0 and y = 2 into the equation kx + 3y = 6, we get k(0) + 3(2) = 6
Notice that, since x = 0, the value of k is irrelevant.
So, (0, 2) will ALWAYS be a point on the line kx + 3y = 6

Cheers,
Brent
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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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Here for any value of k points should lie on the line means we can substitute any value for k so let k =1,2 then the equations will be L
x+3y=6 --------(1)
and
2x+3y=6 ---------(2)
Now after solving the simultaneous equations above we would get the values for x as 0 and y as 2.
Hence (0,2) Option B
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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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kx + 3y = 6

K * 0 + 3 * 2 = 6
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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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Bunuel wrote:
In the xy-coordinate plane, which of the following points must lie on the line kx + 3y = 6 for every possible value of k?

(A) (1,1)
(B) (0,2)
(C) (2,0)
(D) (3,6)
(E) (6,3)

Kudos for a correct solution.

Kx+3y = 6

since we need to make the coordinates independent of K - Lets put x = 0.
3y = 6 so y =2

We get a point (0,2) so B.
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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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Bunuel wrote:
In the xy-coordinate plane, which of the following points must lie on the line kx + 3y = 6 for every possible value of k?

(A) (1,1)
(B) (0,2)
(C) (2,0)
(D) (3,6)
(E) (6,3)

Kudos for a correct solution.

Property : The points must satisfy the given equation of line in order to lie on the line

kx + 3y = 6 will be satisfied only when x co-ordinate is zero as k is unknown variable.

Hence, Without checking any option the correct answer must have x-cp-ordinate zero which is the case in option B

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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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Although I took >3 mins to solve this the first time, after some thinking, I could see one nice takeaway from this question - for any line y=mx+c, the y intercept - c - is a point that will always exist on the line, independent of x or slope m. In this case, the y-intercept is (0,2). So with any value of k or x, (0,2) will always be a point on this line and therefore is the correct answer.

Similarly, let's say the equation of the line was 6x + ky = 6, try thinking what will be a point that will always lie on this line for every possible value of k?
(1,0), i.e. the x-intercept

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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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We're looking for a set of coordinates that are not affected by the value of k. in answer choice B (0,2), the x-coordinate is 0 and y = 2
So, if we plug x = 0 and y = 2 into the equation kx + 3y = 6, we get k(0) + 3(2) = 6
Notice that, since x = 0, the value of k is irrelevant.
So, (0, 2) will ALWAYS be a point on the line kx + 3y = 6
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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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Attached is a visual that should help.
Attachments Screen Shot 2016-05-10 at 5.53.01 PM.png [ 80.64 KiB | Viewed 24059 times ]

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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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Bunuel wrote:
In the xy-coordinate plane, which of the following points must lie on the line kx + 3y = 6 for every possible value of k?

(A) (1,1)
(B) (0,2)
(C) (2,0)
(D) (3,6)
(E) (6,3)

Kudos for a correct solution.

Yes all the above methods are correct but we are forgetting one basic fact.
Form eqn of line we have y = -k/3*x+2.
From eqn we can clearly see that the y intercept of line = 2. So no matter what slope the line has intercept of y-axis will always be 2.
So point (0,2) will always be on the line.

Ans is B
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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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Bunuel wrote:
In the xy-coordinate plane, which of the following points must lie on the line kx + 3y = 6 for every possible value of k?

(A) (1,1)
(B) (0,2)
(C) (2,0)
(D) (3,6)
(E) (6,3)

Kudos for a correct solution.

Here, my job is to remove 'K' from the equation.
The equation says:
kx + 3y = 6
If i want to remove 'k' from the equation, then i must be put x=0. So, B (0,2) makes sense.
kx + 3y = 6
---> k*0+3*2=6
---> 0+6=6
---> 6=6, whic is always correct
So, the correct answer is B.
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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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I did this by substituting each of the values. For (0,2) any value of k still gives kx =0. therefore (0,2)
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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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I changed the given equation to the slope y-intercept form;

y=(-k/3)x+2,

Since in this form, the constant is always the Y-intercept i expect the answer to be (something,2). Thus the answer is B. Is this correct?
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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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Put the given equation in y=mx+b form, as already indicated above.
Now test the answers till you find the match that LHS = RHS, only option B yields 2=2, in which also k gets irrelevant as you multiply k by 0!
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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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Bunuel wrote:
In the xy-coordinate plane, which of the following points must lie on the line kx + 3y = 6 for every possible value of k?

(A) (1,1)
(B) (0,2)
(C) (2,0)
(D) (3,6)
(E) (6,3)

Kudos for a correct solution.

1. The point must lie on the line kx + 3y = 6
2. Every possible value of K must satisfy

So when will every possible value of K satisfy this equation? When the value of K does not impact the solution and this will happen when x = 0.
Only Option B has X = 0. Also, because the point must lie on the line, y has to be equal to 2 when x = 0.
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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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Bunuel wrote:
In the xy-coordinate plane, which of the following points must lie on the line kx + 3y = 6 for every possible value of k?

(A) (1,1)
(B) (0,2)
(C) (2,0)
(D) (3,6)
(E) (6,3)

Kudos for a correct solution.

Hi niks18 here is my solution which i dont understand myself

so we have: $$kx-3y = 6$$

now i will rewrite in the form of y-intercept formula $$y = mx+b$$

$$3y = 6-kx$$

$$3y = -kx+6$$

$$y = \frac{-kx}{3}+ 2$$

I have no idea what -kx is here i got stuck, how to answer this question gimme a hint Retired Moderator D
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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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dave13 wrote:
Bunuel wrote:
In the xy-coordinate plane, which of the following points must lie on the line kx + 3y = 6 for every possible value of k?

(A) (1,1)
(B) (0,2)
(C) (2,0)
(D) (3,6)
(E) (6,3)

Kudos for a correct solution.

Hi niks18 here is my solution which i dont understand myself

so we have: $$kx-3y = 6$$

now i will rewrite in the form of y-intercept formula $$y = mx+b$$

$$3y = 6-kx$$

$$3y = -kx+6$$

$$y = \frac{-kx}{3}+ 2$$

I have no idea what -kx is here i got stuck, how to answer this question gimme a hint Hi dave13

Read the question carefully. It says “something” is true for any value of k. That is you need to make k irrelevant here. So how will that be possible? This is possible in ONLY one way when you multiply k by 0 because any number multiplied by 0 is 0.

In this question you have kx. So do you have any option that makes x=0?, if yes then that will be your answer.

What you did is you found out the slope of the straight line “m” which comes out to be -k/3. All these workings are not required if you understood the above method.

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Bunuel VeritasPrepKarishma. Could you please show a detailed approach and the theory underneath this problem? Thanks.
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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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Bunuel VeritasPrepKarishma. Could you please show a detailed approach and the theory underneath this problem? Thanks.

The first thing I do here is put it in the form of y = mx + c because that tells me a lot about the line represented by the equation.

y = -kx/3 + 2
Slope = -k/3 and y intercept is 2 (which means the line passes through (0, 2)). So depending on the value of k, we will draw a line through the point (0, 2).

For every value of k means no matter what the slope is, the line must pass through the point we need. We know only one such point (0, 2). Do we have it in the options? Yes.

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Re: In the xy-coordinate plane, which of the following points must lie on  [#permalink]

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Bunuel wrote:
In the xy-coordinate plane, which of the following points must lie on the line kx + 3y = 6 for every possible value of k?

(A) (1,1)
(B) (0,2)
(C) (2,0)
(D) (3,6)
(E) (6,3)

Kudos for a correct solution.

Let's rewrite the equation in slope y-intercept form y = mx + b, where m is the slope of the line and b is the line's y-intercept

Take: kx + 3y = 6
Subtract kx from both sides: 3y = 6 - kx
Divide both sides by 3 to get: y = 2 - kx/3
Rewrite as: y = (k/3)x + 2
So, the slope of the line is k/3 and the y-intercept is 2

If the y-intercept is 2, then the line must pass through the point (0, 2)

Cheers,
Brent
_________________ Re: In the xy-coordinate plane, which of the following points must lie on   [#permalink] 20 Aug 2018, 08:01

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