Solve the simultaneous equations x + y = 3 and x^2 + y^2 = 5

Find an expression for y
y = 3 - x
sub this into the second equation
x^2 + (3 - x)^2 = 5
then expand the brackets
x^2 + x^2 - 6x + 9 = 5
simplify
2(x^2 - 3x + 2) = 0
factorise
2(x - 2)(x - 1) = 0
so x = 2, 1
to find the y value sub into the original equation to get
y = 3 - 1 = 2
y = 3 - 2 = 1
so the solutions are
y = 2, x = 1
and
y = 1, x = 2