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To solve the given problem, we need to determine the equation of the line passing through the points \((0, 7)\) and \((-2, 1)\). We'll solve this by first filling in the matrix form to find the determinant, and then using that to derive the equation of the line.

### Step 1: Matrix Setup for the Determinant

We are provided with the following points:
- Point 1: \((0, 7)\)
- Point 2: \((-2, 1)\)

The general form of the determinant for a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is as follows:

\[
\text{det} \left[ \begin{array}{ccc}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x & y & 1 \\
\end{array} \right] = 0
\]

### Filling in the Matrix:

Using the given points, the matrix becomes:

\[
\text{det} \left[ \begin{array}{ccc}
0 & 7 & 1 \\
-2 & 1 & 1 \\
x & y & 1 \\
\end{array} \right] = 0
\]

To compute the determinant, we use the cofactor expansion:

\[
\text{det} = 0 \cdot \text{det}\left( \begin{array}{cc} 1 & 1 \\ y & 1 \end{array} \right) - 7 \cdot \text{det}\left( \begin{array}{cc} -2 & 1 \\ x & 1 \end{array} \right) + 1 \cdot \text{det}\left( \begin{array}{cc} -2 & 1 \\ x & y \end{array} \right)
\]

This simplifies to:

\[
\text{det} = -7 \cdot ( (-2 \cdot 1) - (x \cdot 1)) + ( (-2 \cdot y) - (x \cdot 1))
\]
\[
\text{det} = -7 \cdot (-2 - x) + (-2y - x)
\]
\[
\text{det} = 14 + 7x - 2y - x = 0
\]

Simplifying further:

\[
6x - 2y + 14 = 0
\]

### Step 2: Solve for the Equation of the Line

Now, we simplify the above equation:

\[
6x - 2y = -14
\]

Dividing the entire equation by 2:

\[
3x - y = -7
\]

Rearranging to the slope-intercept form (\(y = mx + b\)):

\[
y = 3x + 7
\]

Thus, the equation of the line is:

\[
y = 3x + 7
\]

### Final Answers:

1. The matrix to fill in is:
\[
\text{det} \left[ \begin{array}{ccc}
0 & 7 & 1 \\
-2 & 1 & 1 \\
x & y & 1 \\
\end{array} \right] = 0
\]

2. The equation of the line is \(y = 3x + 7\).

Would you like further details or have any questions?

Here are five related questions to expand on this topic:
1. How do you calculate the determinant of a 3x3 matrix in general?
2. What is the significance of the determinant being zero in this context?
3. How does the slope formula relate to the determinant method for finding a line?
4. How would the equation change if the two points were different?
5. Can you explain how to convert any linear equation to slope-intercept form?

**Tip:** Always check your matrix setup carefully, as one misplaced value can lead to a wrong determinant result.

Fill in the matrix in the equation below to find the line that passes through points (0, 7) and (-2, 1). What is the equation of the line that passes through points (0, 7) and (-2, 1)?

Learn how to find the equation of a line passing through two points using matrix determinants. This detailed example covers finding the line equation through points (0, 7) and (-2, 1) with step-by-step explanations. Ideal for high school students studying linear algebra.

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