After simplification we find β1 + β3E = 0 or E = −β1/β3. Hence the first two lines cross at E = −β1/β3. Similarly, by equating the second and third lines we can see that they also cross at the same point. Accordingly, the traditional parameterization of the interaction effect implies that all lines cross at the same point or all must be parallel depending on β3 coefficient. There is no a priori reason to assume this would be the case or to force this constraint when modeling interaction.
