Prove the following equations:

$x+x^{\text{'}}y=x+y$

$x+xy=x$

$x^{\text{'}}yz+x^{\text{'}}y{z}^{\text{'}}+xz=x^{\text{'}}y+xz$

$xy+x^{\text{'}}z+yz=xy+x^{\text{'}}z$

$(A+B)(A+C)=AC+A^{\text{'}}B$

$A^{\text{'}}B+B^{\text{'}}{C}^{\text{'}}+AB+B^{\text{'}}C=1$