# Multiple regression model from factorial experiment

Given the following factorial experiment, how does one construct an 'equivalent' multiple regression model? This appeared in an exam and left me almost clueless.

\begin{align*} \mu_{11} &= \mu + \alpha_1 + \gamma_1 + \theta_{11}\\ \mu_{12} &= \mu + \alpha_1 + \gamma_2 + \theta_{12}\\ \mu_{21} &= \mu + \alpha_2 + \gamma_1 + \theta_{21}\\ \mu_{22} &= \mu + \alpha_2 + \gamma_2 + \theta_{22}\\ \mu_{21} &= \mu + \alpha_2 + \gamma_1 + \theta_{21}\\ \mu_{31} &= \mu + \alpha_3 + \gamma_1 \\ \mu_{32} &= \mu + \alpha_3 + \gamma_2 \end{align*}

The problem essentially extends to say: 'Assume $\mu=1, \alpha_1=2, \alpha_2=-3, \gamma_1=5, \theta_{11}=4$ find rest of parameters and regression coefficients'

My first attempt was to treat $(\mu, \alpha_1, \alpha_2, \alpha_3, \gamma_1, \gamma_2, \theta_{11}, \theta_{12}, \theta_{21}, \theta_{22})$ as my $b$ vector in : $$y = Xb$$

where $y=(\mu_{11} \dots \mu_{32})_{6 \times 1}$ and $X$ being the design matrix. But then how would I estimate the 'rest of parameters' if the values of $y$ vector are unknown(except $u_{11}$)