When we have several explanatory variables $x_j$ and a variable $y$ to be predicted the linear model becomes
\[
y_i=\beta_1x_{i,1}+\beta_2x_{i,2}+ \dots + \beta_px_{i,p}+\varepsilon_i \quad i=1,\dots,n
\]
or
\[
\mathbf{y}=\mathbf{X\beta}+\mathbf{\varepsilon}.
\]
The OLS solution
\[
\min_{\mathbf{\beta}}\| \mathbf{y}-\mathbf{X\beta}\|^2=\min_{\mathbf{\beta_1,\dots,\beta_p}}\sum_{i=1}^n(y_i-\sum_{j=1}^p(\beta_{j}x_{i,j}))^2
\]
is
\[
\mathbf{\hat{\beta}}=(\mathbf{X'X)^{-1}X'y}.
\]
\]