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Find k if the systemhas exactly one solution in real numbers.

 May 8, 2020
 #1
avatar+21957 
+2

The original equation (the absolute value equation) can be graphed in parts:

 

If  x  <=  -23, then the equation is:    y  =  -(x + 23) - (x - 5) - (x - 48)  =  -3x + 30

If  -23  <=  x  < = 5, the equation is:  y  =  (x _ 23) - (x - 5) - (x - 48)  =  -x + 76

If  5  <= x  <= 48, the equation is:     y  =  (x + 23) + (x - 5) - (x - 48)  =  x + 66

if  x  >=  58, then the equation is:      y  =  (x + 23) + (x - 5) + (x - 48)  =  3x - 30

 

If you graph these, you will see that a line with a slope of 2 will intersect this graph in one point only if the

point is the intersection of the two parts:  y  =  x + 66  and  y  =  3x - 30  which occurs at  (48, 114).

 

For  y  =  2x + k  --->   114  =  2(48) + k   --->  k  =  18.

 May 9, 2020

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