How many integral values of k are possible, if the lines 3x+

I got -0.5 < k < 4.5

hence the integral values are 0,1,2,3,4. should be B. Please check. I am not a fan of algebra, sorry

Can you explain how you got this range?

@gmat1220
But now you got the correct range and correct answer .

I was not taking values of x and y as independent. Thats why i got confused.
Thanks for the explanation.

Man this was monster ! especially my equations go completely haywire when I look at algebra.

Can someone pls explain how to arrive at these 2 equations mentioned above.
x = -18 - 36k / (4k^2 + 9)
y = (108 - 24k) / 4 (4k^2 + 9)

The given equations are

3x + 4ky = -6 ------1
kx - 3y = -9 ------2

Try solving the two equations:

1. Multiply eq 1 by k, 3kx + 4k^2y = -6k ---------3
2. Multiply eq 2 by 3, 3kx - 9y = -27 ---------4

Solving eq, 3 and 4, we can get the value of y.

where y = (27-6k)/(4k^2 + 9)..

Since y should be > 0, 27-6k > 0, solving k<4.5

Similarly, solving for x gives or sub y in eq 2 , x = -6(6k+3) / 4k^2 + 9

since x < 0, 6k+3 should be > 0 => 6k+3 > 0, solving k < -0.5

Hence k has 5 Integral values 0, 1, 2, 3 and 4.

I have a question..
when we get- y=27-6k/4k^2+9 and since y>0 then why do we assume just- 27-6k>0 instead of 27-6k>0 and 4k^2+9>0 OR 27-6k<0 and 4k^2+9<0
I know I'm missing something here but I can't understand what..Please help!

The value 4k^2+9>0 will always be positive its the numerator which will make the diff

Posted from my mobile device

The solution provided by beyondgmatscore is perfect. A student asked me to add a few steps, so that what I'll do.

GIVEN:
3x + 4ky + 6 = 0
kx - 3y + 9 = 0

Rewrite as:
3x + 4ky = -6
kx - 3y = -9

NOTE: The intersection of the two lines will be at the point where the same pair of x- and y-values satisfy BOTH equations.
In other words, we want to SOLVE the system of equations above.

So, let's use the ELIMINATION method to solve the system.

First we'll eliminate the x terms
So, multiply both sides of the TOP equation by k, and multiply both sides of the BOTTOM equation by 3 to get:
3kx + 4k²y = -6k
3kx - 9y = -27

Subtract the bottom equation from the top equation to get: 4k²y + 9y = -6k + 27
Rewrite as: y(4k² + 9) = 27 - 6k
Divide both sides by (4k² + 9) to get: y = (27 - 6k)/(4k² + 9)

KEY CONCEPT: If the two lines intersect in the 2nd quadrant, then the x-value must be NEGATIVE, and the y-value must be POSITIVE

If the y-value must be POSITIVE, we can write: (27 - 6k)/(4k² + 9) > 0
Since (4k² + 9) is POSITIVE for all values of k, we can conclude that 27 - 6k is POSITIVE
In other words: 27 - 6k > 0
Add 6k to both sides to get: 27 > 6k
Divide both sides by 6 to get: 4.5 > k
--------------------------------------

Now we'll perform similar steps to isolate x (by eliminating the y terms)

Take:
3x + 4ky = -6
kx - 3y = -9

Multiply both sides of the TOP equation by 3, and multiply both sides of the BOTTOM equation by 4k to get:
9x + 12ky = -18
4k²x - 12ky = -36k

Add the two equations to get: 9x + 4k²x = -18 - 36k
Factor left side: x(9 + 4k²) = -18 - 36k
Divide both sides by (9 + 4k²) to get: x = (-18 - 36k)/(9 + 4k²)

KEY CONCEPT: If the two lines intersect in the 2nd quadrant, then the x-value must be NEGATIVE, and the y-value must be POSITIVE

If the x-value must be NEGATIVE, we can write: (-18 - 36k)/(9 + 4k²) < 0
Since (9 + 4k²) is POSITIVE for all values of k, we can conclude that -18 - 36k is NEGATIVE
In other words: -18 - 36k < 0
Add 18 to both sides to get: -36k < 18
Divide both sides by -36 to get: k > -0.5
--------------------------------------

Now combine both results to get: -0.5 < k < 45

How many integral values of k are possible?
If -0.5 < k < 45, then the possible INTEGER values of k are: 0, 1, 2, 3, 4 (5 possible values)

Answer: A

Cheers,
Brent

solve both quadratic equations
x = (6k+ 1)(-6) / (4k2 +9)

y = (27-6k) / (4k2 +9)

He says that they intersect in 2nd quadrant.

Hence, x < 0, y >0 ( Any value in second quadrant)
x <0
-6(1+6k) < 0
k> -1/6
K>-0.166 ----1

y>0
27-6k > 0
6k < 27
k < 27/6
k < 4.5 ---2

Hence from 1 & 2
-0.166< K < 4.5

Integral values K can take = 0,1,2,3,4
Answer is 5 option A.

Do these kinds of question occur only in GMAT or they can also be shown up in GRE?

With the given equation we have to individually figure out the values
x = -18 - 36k / (4k^2 + 9)
y = (108 - 24k) / 4 (4k^2 + 9)

Hence -18 - 36 k < 0
So k > -0.5

And 108 - 24k >0
k < 108 / 24
k < 18 / 4
k < 9/2
k < 4.5

-0.5 < k < 4.5
Hence IMO 5

GMAT doesn't give you such tedious questions.

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