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# For what values of k will the pair of equations 3x + 4y = 12 and kx +

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Senior Manager
Joined: 12 Jun 2006
Posts: 397
For what values of k will the pair of equations 3x + 4y = 12 and kx +  [#permalink]

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13 Apr 2007, 00:22
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45% (medium)

Question Stats:

66% (01:35) correct 34% (02:11) wrong based on 534 sessions

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For what values of k will the pair of equations 3x + 4y = 12 and kx + 12y = 30 does not have a unique solution?

A. 12
B. 9
C. 3
D. 7.5
E. 2.5
Director
Joined: 14 Jan 2007
Posts: 502
Re: For what values of k will the pair of equations 3x + 4y = 12 and kx +  [#permalink]

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13 Apr 2007, 02:51
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A system of linear equations with a unique solution is characterized by the graphs of two lines whose intersection is a single point, and the coordinates of this point satisfy both equations.
So we have to find out the the value K so that the lines become parallel. Parallel lines don't intersect each other. That's why having no unique solution.

Two parallel lines have their slopes equal.
Slope of the line 3x+4y = 12, = -3/4

Putting K =9 in the line kx+12y = 30 , slope of this line = -3/4

Hence for k =9 the second line is parallel to the first one. Hence the lines will not have a unique solution.
##### General Discussion
Intern
Joined: 06 May 2016
Posts: 14
WE: Education (Education)
Re: For what values of k will the pair of equations 3x + 4y = 12 and kx +  [#permalink]

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16 May 2016, 01:24
3
To solve two variables we need two distinct equations.

If X and Y are in the same ratio in both equations we cannot solve it.

Y1:Y2
4:12
1:3

So if X1:X2 is in the ratio 1:3 there is no distinct equation
1:3
3:9

(B) If X = 9 there is no solution.
VP
Joined: 01 May 2006
Posts: 1431
Re: For what values of k will the pair of equations 3x + 4y = 12 and kx +  [#permalink]

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13 Apr 2007, 03:00
1
1
(B) as well... I agree with what vshaunak@gmail.com said above

For 2 equations of line, we have 3 possible cases:
> 1 point intersect the 2 lines = 2 different slopes for the lines
> No intersection = Same slopes and different Y-interceptors
> Infinite number of interesections = Same slopes and same Y-interceptors

So, the question just asks us for which value of k, the slopes are equal.

o 3x+4y = 12
<=> y = -3/4 * x + 12

o kx+12y = 30
<=> y = -k/12 * x + 30/12

By equalizing the 2 slopes, we have:
-k/12 = -3/4
<=> k = 12*3/4 = 9
Intern
Joined: 13 Oct 2005
Posts: 7
Re: For what values of k will the pair of equations 3x + 4y = 12 and kx +  [#permalink]

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09 Jan 2012, 00:47
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Hi Guys, let me know if this approach is correct(its a shortcut though):)
solving the two eq simultaneously, we get kx-9x=18
so for any value except 9 for K we ill have a unique solution. so 9 is our answer.
CEO
Joined: 07 Jul 2004
Posts: 2917
Location: Singapore
Re: For what values of k will the pair of equations 3x + 4y = 12 and kx +  [#permalink]

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13 Apr 2007, 06:08
1
2x+4y = 12 --> 4y = 12-2x --> y = -0.5x+3
kx+12y = 30 --> 12y = -kx+30 --> y = -kx/12 +2.5

The pairs will not have a unique solution if the lines are parallel. To find this, just make sure both lines have similar slopes. Above, I have place the equations in the standard straight-line equation y = mx+c where m is the gradient of the line and c is the y-intercept.

so since y = -0.5x+3 has a gradient of -0.5, then we must make sure y = kx/12+2.5 has a gradient of -0.5 too. To have this, we need k = 9.
Manager
Joined: 14 Jan 2016
Posts: 82
Location: India
Concentration: Marketing, General Management
GMAT 1: 710 Q50 V35
GMAT 2: 750 Q50 V41
Re: For what values of k will the pair of equations 3x + 4y = 12 and kx +  [#permalink]

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18 Jan 2016, 07:45
A linear equation in two variables may have : 1 solution(intersecting lines), infinite solution(overlapping lines) or no solution (parallel lines)

For a system of equations to not have a unique solution, there should either be no solution or infinite solutions.

ax+by= c
px+qy=r

These will have no solution when :-
(a/p) = (b/q) =/= (c/r)

These will have infinite solutions when :-
(a/p) = (b/q) = (c/r)

As 4/12 =/= 12/30, thus the system should not have any solution.
4/12 = 1/3 thus 3/k = 1/3
k = 3*3 = 9.
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Intern
Joined: 09 Aug 2015
Posts: 4
Re: For what values of k will the pair of equations 3x + 4y = 12 and kx +  [#permalink]

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15 May 2016, 22:45
1
i approached this problem in slightly different way,

we have 2 equations
1. 3x + 4y = 12 -- > 9x + 12y = 36

2. kx + 12y = 30

substract 1-2, we get (9-K)x = 6

i.e. x = 6/(9-k)

then, by looking at options, we get some value of x except for B. When we put k=9, x becomes 6/0 and hence answer is B
Intern
Joined: 06 Dec 2014
Posts: 8
Re: For what values of k will the pair of equations 3x + 4y = 12 and kx +  [#permalink]

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19 Oct 2016, 05:59
This can also be approached this way.

Multiply 3x+4y=12 with 3 and replace 12 y with 30-kx.

This gives us x(9-k)-6. Hence, x= 6/(9-k). We get value for x for everything besides k=9 as that makes the denominator 0.

Hope this helps
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Joined: 09 Sep 2013
Posts: 15024
Re: For what values of k will the pair of equations 3x + 4y = 12 and kx +  [#permalink]

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14 Jan 2020, 02:56
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Re: For what values of k will the pair of equations 3x + 4y = 12 and kx +   [#permalink] 14 Jan 2020, 02:56