If $$(\alpha, \beta)$$ and $$(\gamma, \beta)$$ are the roots of the simultaneous equations $$|x-1|+|y-5|=1$$, $$y = 5+|x-1|$$, then the value of $$\alpha + \beta + \gamma$$ is

We are given the equations: $$|x-1|+|y-5|=1$$, $$y = 5+|x-1|$$

From the second equation, we get:

$$|x-1|=y-5$$

Substituting this into the first equation

$$|x-1|+|y-5|=1$$

$$y-5+|y-5|=1$$

Now, we will have to make cases

CASE 1: $$y\geq5$$

$$y-5+y-5=1$$

or, $$y=\dfrac{11}{2}$$

Then, $$|x-1|=y-5=\dfrac{1}{2}$$

Which gives $$x=\dfrac{3}{2}$$ or $$x=\dfrac{1}{2}$$

Hence, we get $$2$$ solutions from case 1 for $$(x,y)$$, which are $$\left(\dfrac{1}{2},\dfrac{11}{2}\right),\left(\dfrac{3}{2},\dfrac{11}{2}\right)$$

CASE 2: $$y<5$$

$$y-5-(y-5)=1$$

or, $$0=1$$

Which is never true

So, we cannot get any answer from here.

The common $$y = \beta = \frac{11}{2}$$. 

Taking $$\alpha = \frac{1}{2}$$ and $$\gamma = \frac{3}{2}$$ (or vice versa), we get:

$$\alpha+\beta+\gamma = 2 + \frac{11}{2} = \frac{15}{2}$$.