1. The problem asks to derive the regression lines for determining the relationship between four variables: $y$, $x_1$, $x_2$, and $x_3$. 2. In multiple linear regression, the general form of the regression equation is: $$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \epsilon$$ where $\beta_0$ is the intercept, $\beta_1$, $\beta_2$, and $\beta_3$ are the coefficients for each independent variable, and $\epsilon$ is the error term. 3. To estimate the coefficients, we use the least squares method which minimizes the sum of squared residuals: $$\min_{\beta_0, \beta_1, \beta_2, \beta_3} \sum_{i=1}^n (y_i - \beta_0 - \beta_1 x_{1i} - \beta_2 x_{2i} - \beta_3 x_{3i})^2$$ 4. The normal equations for the coefficients can be written in matrix form as: $$\mathbf{X}^T \mathbf{X} \boldsymbol{\beta} = \mathbf{X}^T \mathbf{y}$$ where $$\mathbf{X} = \begin{bmatrix} 1 & x_{11} & x_{21} & x_{31} \\ 1 & x_{12} & x_{22} & x_{32} \\ \vdots & \vdots & \vdots & \vdots \\ 1 & x_{1n} & x_{2n} & x_{3n} \end{bmatrix}, \quad \boldsymbol{\beta} = \begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3 \end{bmatrix}, \quad \mathbf{y} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix}$$ 5. Solving for $\boldsymbol{\beta}$ gives: $$\boldsymbol{\beta} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y}$$ 6. This solution provides the estimated regression coefficients that define the regression line relating $y$ to $x_1$, $x_2$, and $x_3$. 7. In summary, the regression line is: $$y = \hat{\beta}_0 + \hat{\beta}_1 x_1 + \hat{\beta}_2 x_2 + \hat{\beta}_3 x_3$$ where the $\hat{\beta}$ values are estimated from the data using the formula above. This method allows us to understand how each independent variable $x_1$, $x_2$, and $x_3$ influences the dependent variable $y$ while controlling for the others.