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# Find k

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Find k if the system

\begin{align*} y &= |x + 23| + |x - 5| + |x - 48|, \\ y &= 2x + k \end{align*}
has exactly one solution in real numbers.

May 16, 2020

#1
+30921
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The graph below shows that the value of k must lie between 0 and 100.  Clearly, you want the value of k that just clips the red graph at x = 50, so find the value of y on the red graph at x = 50 using the first equation, plug that into the second equation together with x = 50 and rearrange the resulting expression to find k.

Edit:  The value of x = 50 is incorrect - see below.

May 16, 2020
edited by Alan  May 17, 2020
#2
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Hi Alan, thanks for responding. I tried your method as follows:

Plugging x=50 into the first equation:

y= 73+45+2

y=120

Plugging x=50 and y=120 into the second equation and solving for k:

120=2(50) + k

k=20

I tried inputting this answer but it is saying my answer is wrong. I also don't understand where you got x=50 from.

Guest May 16, 2020
#3
+30921
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Oops!  I was careless in looking at the graph and misled myself (and you!).   The point where the two functions meet at a single point is at x = 48, not 50.  Plug this into the first function to get y (= 114).  Then use this value of y in the second function, together with x = 48 to get k ( =  18).

Alan  May 17, 2020
edited by Alan  May 17, 2020
#4
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Oh that makes sense, thank you!

Guest May 17, 2020