… did [the OP] intend to write a+b+c = 2
and separately (a^2 + b^2 + c^2) = 4
Not sure, Cubist, but those two equations have infinite Real solutions. I visualized treating c as a parameter and subtracting it from both sides. Then we'd have in the second equation a circle of radius r=(4-c^2) which could be centered on horizontal a-axis and vertical b-axis, and the first equation would be an intersecting line b=-a+(c-2) -- that is, a line with slope -1 and b-intercept (c-2).
As c increases, the radius 4-c^2 decreases toward zero, and the b-intercept of the line rises. As long as c is set within (0,2), I'm thinking the line will intersect in two places, giving infinite pairs of solutions. At first, I thought c could be 0 or 2 also, but I'd forgotten that we can't have zero-values for a, b or c (because they are denominators in the rest of the exercise).
When I tried extending my visualization into negative values of c, the back of my eyes started to ache, so I stopped. Clearly, the circle's size would increase and the line would drop.
Now, my puzzlement is: Given a generalized solution for a, b and c, how would we find expressions for x, y and z when we have only one equation? Maybe a shortcut belonging to a specific interpretation of the givens is needed, including the expression we're supposed to deduce (xy+zy+xz). Or, maybe the OP created this, while working on something we can't see.
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