Confusion of “slope” and “intercept” in linear regression

Question

I read John Fox's book "Regression diagnostics: An introduction". In Chapter 2, he gave an example which I quoted below:

… if $y$ is income, $x$ is education in years, and $g$ is the factor
gender, with levels male, female, and nonbinary, coding the dummy
regressors $d_f = 1$ for females and 0 otherwise, and $d_m = 1$ for
males and 0 otherwise,

the model $y = \beta_o + \beta_1x + \beta_2d_f + \beta_3d_m + \epsilon$ assumes the same education slope for all three genders but
potentially different intercepts.

Below is what I do not understand:

Why does this model have the "same slope" but "different intercepts"? I thought their slopes are different with different betas ($\beta_1, \beta_2, \beta_3$). On the contrary, the intercept is the same ($\beta_0$). Anyway, why the intercept can be different (there is only one variable $\beta_0$)?

Answer 1

For a male person, the equation is $$y=\beta_1x + (\beta_0+\beta_3)+\epsilon$$ and for a female person, the equation is $$y=\beta_1x + (\beta_0+\beta_2)+\epsilon$$

These two equations have the same education slope and potentially different intercepts as described in the text.