Prove $Re (z) = \frac{z + z^* }{2}$ and $Im (z) = \frac{z − z^*}{2i}$ In texts on complex numbers I often see an exercise that asks to prove the following:

$$Re (z) = \frac{z + z^*}{2}$$
$$Im (z) = \frac{z − z^*}{2i}$$
where $z = x + iy$ and $z^* = x - iy$

I understand the meaning of complex numbers, but can't seem to find a path to proving these two identities.
Any insight would be appreciated.

Solve the following equations in $x,y$:
$z = x + iy \cdots(1)$
and $z^* = x - iy \cdots (2)$
Adding $1$ and $2$ we get  $z + z^* = 2x \implies x = \frac{z + z^*}{2}$ i.e. $Re (z) = (z + z^* )/2 $
Subtracting $1$ and $2$ we get  $z - z^* = 2iy \implies y = \frac{z - z^*}{2i}$ i.e. $Im (z) = (z − z^*)/2i$
