The teacher and writer is coming to the house.
此处用is是因为主语the teacher and writer是一个人,也就是说,作主语的这个人既是老师,又是作家,本质上是一个人,身兼多职.因此谓语动词用is而不是are.
如果此处主语换成the teacher and the writer则是指两个人,这种情况下才用are.