Typically I see $y$ as the response variable and $x$ as the predictor variable. $y_i=\beta_0+\beta_1x_i+\epsilon_i,\quad i=1,\dots,n.$$
What other forms can the equation have? Would it be valid if we omit the $\epsilon$, for example: $y_i=\beta_0+\beta_1x_i+β_2x_i?$
What if we don't include all $\beta$'s and $x$'s? Or should we raise them to some power: $y_i=\beta_0+\beta_1x_i+ \beta_2*(x^2) + \beta5*(x^5)+\epsilon?$