The given notation appears to describe a determinant of a 3x3 matrix, and you're likely looking to solve for the lines that pass through particular points. The matrix determinant often corresponds to the condition for collinearity when three points are provided. Let's break it down:

You seem to have:

x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{matrix} \right| = 0$$ This determinant form is used to check if three points $$(x_1, y_1)$$, $$(x_2, y_2)$$, and $$(x_3, y_3)$$ are collinear. In your case, the matrix seems to be: $$\left| \begin{matrix} 2 & 4 & 1 \\ y & 8 & 1 \\ 1 & 0 & 1 \end{matrix} \right| = 0$$ Now, let's expand the determinant: $$\left| \begin{matrix} 2 & 4 & 1 \\ y & 8 & 1 \\ 1 & 0 & 1 \end{matrix} \right| = 2 \left( 8 \cdot 1 - 0 \cdot 1 \right) - 4 \left( y \cdot 1 - 1 \cdot 1 \right) + 1 \left( y \cdot 0 - 1 \cdot 8 \right)$$ This simplifies to: $$2(8) - 4(y - 1) - 8 = 0$$ Simplifying further: $$16 - 4(y - 1) - 8 = 0$$ $$16 - 4y + 4 - 8 = 0$$ $$12 - 4y = 0$$ $$4y = 12$$ $$y = 3$$ Thus, the value of $$y$$ is **3**. The lines pass through the points $$(2, 4)$$, $$(3, 8)$$, and $$(1, 0)$$. Would you like more details on how this determinant method works or any other clarifications? Here are 5 related questions you might find helpful: 1. How do you calculate the determinant of a 3x3 matrix? 2. What is the significance of collinearity in a determinant-based approach? 3. How can you find the equation of a line passing through two points? 4. How do you generalize this method for n-dimensional geometry? 5. How do determinants apply to other geometric shapes? **Tip:** When solving determinants, simplifying the matrix row or column with zeros can save computation time.