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.

Guest May 16, 2020

#1**+3 **

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.

Alan May 16, 2020

#2**0 **

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**+1 **

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