In the xy-coordinate plane, which of the following points must lie on

kx + 3y = 6

Answer B (0,2)
K * 0 + 3 * 2 = 6

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.

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

Answer: Option B

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?


HTH

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
so my answer is b

Attached is a visual that should help.

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

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.

Answer (B)

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)

Answer: B

Cheers,
Brent

Hi All,

With this prompt, we’re asked which of the following points will ALWAYS lie on the line (K)(X) + 3(Y) = 6 for EVERY possible value of K. This is a great ‘concept question’, meaning that you don’t have to do a lot of math to answer it if you recognize the concept involved (and that concept is a Number Property Rule that you probably already know…).

To start, if the value of K can vary – and the sum will always be 6 - then the relative values of X and/or Y will almost always change based on the value of K. From the way the question is worded though, we know that one of the co-ordinates will ALWAYS appear on the line. Thus, we have to look for an answer that ‘counters’ the fact that K could vary. If you don’t immediately see it, then you can TEST THE ANSWERS to prove it…

Answer A: (1,1)… With that co-ordinate, the equation would become…
K(1) + (3)(1) = 6…. So K can ONLY equal 3. This does NOT match what we were told.
Eliminate Answer A

Answer B: (0,2)… With that co-ordinate, the equation would become…
K(0) + (3)(2) = 6…. Since (3)(2) = 6, then and K(0) will ALWAYS equal 0, we have a correct equation and the value of K is IRRELEVANT (meaning that if K changes, the point will still appear on the line.

Final Answer:

GMAT Assassins aren’t born, they’re made,
Rich

Solution:

To "must lie" for "every value of k", on the line the k term has to be removed.

This is possible if x = 0 as that would make k x =0 and the equation is k independent.

(option b) of (0,2) when plugged in suffices the condition. (option b)

Devmitra Sen
GMAT SME

Solution:

Let’s express this equation in slope-intercept form by isolating y:

3y = -kx + 6

y = -k/3 x + 2

We see that this line has a y-intercept of 2, regardless of what the slope -k/3 is (hence regardless of the value of k as well). Since the ordered pair (0, 2) is in fact the y-intercept, we see, then, that the point (0, 2) must lie on the line.

Answer: B

Answer: Option B

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To determine which point must lie on the line kx + 3y = 6 for every possible value of k, we can substitute each point into the equation and check if the equation holds true.

Let's substitute the points into the equation:

A. (1,1): k(1) + 3(1) = k + 3 ≠ 6 (not true for every possible value of k)
B. (0,2): k(0) + 3(2) = 0 + 6 = 6 (true for every possible value of k)
C. (2,0): k(2) + 3(0) = 2k + 0 = 2k ≠ 6 (not true for every possible value of k)
D. (3,6): k(3) + 3(6) = 3k + 18 ≠ 6 (not true for every possible value of k)
E. (6,3): k(6) + 3(3) = 6k + 9 ≠ 6 (not true for every possible value of k)

From the above calculations, we can see that only point (0,2) satisfies the equation kx + 3y = 6 for every possible value of k.

Therefore, the correct answer is (B) (0,2).