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# Multiple values of K

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For which values of $k$ does the equation $\frac{x-1}{x-2} = \frac{x-k}{x-6}$ have no solution for $x$? Enter all the possible values of $k,$ separated by commas.

Dec 26, 2020

#1
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(By the way I got that one value of k is 5)

Dec 26, 2020
#2
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That's right, k = 5 is the only solution.

Guest Dec 26, 2020
#3
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No, I tried 5, it is incorrect, there must me more.

Dec 27, 2020
#4
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Wait I thought of another one, if you use k = 6, then the equation will be x-1=x-2, that means -1=-2, which has no solutions, so k= 5, 6, and Idk if there are more

Dec 27, 2020
#5
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x - 1          x  -  k

____  =   _______

x - 2          x  - 6

Cross-multiply

x^2  - 7x  + 6  = x^2 - 2x -kx + 2k

When k  = 5  we  have no solutions because  we will end up with

6 =10    which is impossible

Also...when k  = 6  the rational  function on the  right will  have the graph of the  line  y = 1  (with a "hole" at x = 6)

But the rational  function on the left will have  a horizontal asymptote at y =1.....so  these functions will never intersect when k  = 6

So....no solutions when k= 5 or k  = 6

Dec 27, 2020