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

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For what values of 'k' will the pair of equations 3x+4y = 12 [#permalink] New post 13 Apr 2007, 00:22
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For what values of 'k' will the pair of equations 3x+4y = 12 and kx+12y = 30 not have a unique solution?

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Director
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 [#permalink] New post 13 Apr 2007, 02:51
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.
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 [#permalink] New post 13 Apr 2007, 03:00
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(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
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 [#permalink] New post 13 Apr 2007, 06:08
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.
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Re: For what values of 'k' will the pair of equations 3x+4y = 12 [#permalink] New post 09 Jan 2012, 00:47
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.
Re: For what values of 'k' will the pair of equations 3x+4y = 12   [#permalink] 09 Jan 2012, 00:47
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