This is how I'd go about it, be aware that it's a bit cumbersome and I'm sure someone else will be able to show you another way of going about it.

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!