Fitting multiple line

Not knowing the data in detail, this look like a linear model with "dummys".

A standard linear model looks like ($beta_0$ is the intercept, $beta_1$ is the slope):

$$ y = beta_0 + beta_1 x + u.$$

Now, when you have two distinct "groups" for which there is a "flag" in the data, you can assign a indicator variable or "dummy" (say $d$, a vector with =1 "true" or =0 otherwise). You can add this to your linear model:

$$ y = beta_0 + beta_1 x + beta_2 d + u. $$

$d$ introduces a separate intercept for the assigned group ($d=1$).

You can also add "interaction terms" to allow for a separate slope for group $d$ by simply multiplying $x$ and $d$.

$$ y = beta_0 + beta_1 x + beta_2 d + beta_3 x d + u. $$

Note that since there already is one intercept in the model (the $beta_0$), you can only add "contrasts" to the intercept. So when you have $i$ groups for which you want to have an individual intercept, you would add $i-1$ indicators/dummys to the model. For the "reference group", the intercept will be $beta_0$ and for the group which is identified by the "dummy" ($d$ above), the intercept would be $beta_0 + d$.