Given the equations:
x+y=a
x+z=b
y+z=c
where a, b and c are constants, we want to solve for x, y and z in terms of a, b and c.

It turns out that x=½(a+b-c)
Can you now use intuition without any algebraic manipulation to immediately give the values of y and z in terms of a, b and c?

It is all about symmetry. For example, do you see why a and b have the same coefficient in the expression for x? x is indifferent to y and z. The first equation has x and y and the second has an identical expression for x and z. From the point of view of x, a and b are equivalent. The third equation is y+z=c, which again from x’s point of view, fails to distinguish y from z, but provides different type of information from x’s viewpoint, from the first two equations.