label x2 + x3 = b1 as equation 1
x1 + x3 = b2 as equation 2
x1 + x2 = b3 as equation 3.
You can start with any equation, I'll start with 1. rewrite in terms of an x value. I'll use x3.
x3 = b1 - x2
sub into equation two, x1 + b1 - x2 = b2
rewrite interms of an x, I'll use x1.
x1 = b2 - b1 + x2
sub into equation 3.
b2 - b1 + x2 + x2 = b3
rewrite interms of x2
x2 = (1/2)(b3 - b2 + b1)
That's one of your answers, now sub that back into equation 1.
(1/2)(b3 - b2 + b1) + x3 = b1
rewrite for x3:
x3 = b1 - (1/2)(b3 - b2 + b1)
That's the second of your answers, finally sub this value into equation two.
x1 + b1 - (1/2)(b3 - b2 + b1) = b2
rewrite for x1
x1 = b2 - b1 + (1/2)(b3 - b2 + b1)
Hope that works out right!