How come (x*y)+(x*z')+(y*z) = (x*z')+(y*z)?
You have

$$\bar{x}yz + x\bar{y}\bar{z} + xy\bar{z} + xyz = (x + \bar{x})yz + (y + \bar{y})x\bar{z} = yz + x\bar{z}.$$

Alternatively

$$xy + yz + x\bar{z} =  xy(z + \bar{z}) + yz + x\bar{z} = (1 + x)yz + (1 + y)x\bar{z} = yz + x\bar{z}.$$