i'd be really grateful if someone could complete this simultaneous equation with working out explaining each stage of the process. thanks.

y = 2x + 3

x2 + y2 = 2

You mean this?

\(\displaystyle y\,=\,2x\,+\,3\)

\(\displaystyle x^2\,+\,y^2\,=\,2\)

Plug the first equation into the second where you see the y:

\(\displaystyle x^2 + (2x + 3)^2 = 2\)

Solve for x:

\(\displaystyle x^2 + 4x^2 + 12x + 9 = 2\)

\(\displaystyle 5x^2 + 12x + 7 = 0\)

\(\displaystyle (x + 1)(5x + 7) = 0\)

Set each factor equal to 0 and you'll get

\(\displaystyle x = -1\) or \(\displaystyle x = {-\frac{7}{5}}\)

Plug both values into the first equation to get the corresponding y values.

EDIT: Beaten to it!