https://file.aiquickdraw.com/mathbot/file/17269479056909nfkqf8u.jpg

The task in this image involves two steps: 

1. **Filling in a determinant matrix** to find the line passing through the points \((0,7)\) and \((-2,1)\).
2. **Finding the equation of the line** that passes through these points.

### Step 1: Filling in the Determinant Matrix

The determinant matrix is used to find the equation of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\). The general form of this 2x2 matrix is:

\[
\text{det} \begin{bmatrix} 
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x & y & 1
\end{bmatrix} = 0
\]

For this problem, the points are \((0, 7)\) and \((-2, 1)\). Plugging them into the determinant matrix:

\[
\text{det} \begin{bmatrix} 
0 & 7 & 1 \\
-2 & 1 & 1 \\
x & y & 1
\end{bmatrix} = 0
\]

### Step 2: Finding the Equation of the Line

To find the slope (\(m\)) of the line, use the slope formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 7}{-2 - 0} = \frac{-6}{-2} = 3
\]

Now, using the point-slope form \(y - y_1 = m(x - x_1)\) and the point \((0,7)\):

\[
y - 7 = 3(x - 0) \implies y = 3x + 7
\]

Thus, the equation of the line is \(y = 3x + 7\).

### Summary:
1. The determinant matrix to fill is:

\[
\text{det} \begin{bmatrix} 
0 & 7 & 1 \\
-2 & 1 & 1 \\
x & y & 1
\end{bmatrix} = 0
\]

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

Would you like more details on how determinants are used for finding the equation of a line?

Here are 5 questions to deepen your understanding:
1. What is the meaning of a determinant in a 2D geometric context?
2. How can the slope-intercept form be derived from the point-slope form?
3. What are other methods for finding the equation of a line given two points?
4. How would the equation change if the points were vertical (same \(x\)-coordinates)?
5. How does the determinant test help verify if three points are collinear?

**Tip:** When solving systems or finding equations of lines, organizing the points clearly helps avoid mistakes when filling in determinant matrices.

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

Learn how to find the equation of the line passing through the points (0, 7) and (-2, 1) using a determinant matrix and the slope formula. This problem covers key concepts in linear algebra and coordinate geometry. Suitable for students in Grades 8-10.

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