https://file.aiquickdraw.com/mathbot/file/17255104823086458j4bp.jpg

I will extract and analyze the content from the image to provide a detailed solution. Let's start by reviewing the visible details in the image.

Upon inspection, the image shows a mathematical problem involving:

1. A request to **derive an equation for a line** passing through two points.
2. A **graph** showing the Cartesian plane, with one **diagonal line** plotted.
   
Let me process the information and interpret the relevant details, including the equation or description of the line.The question in the image asks to:

**"Derive an equation for the line passing through the points \((-1, 3)\) and \( (2, -2) \), and graph the line. Also, write the slope of the line."**

Here’s how we approach the solution:

### Step 1: Find the slope of the line

The slope formula \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Plugging in the coordinates \((-1, 3)\) and \( (2, -2) \):

\[
m = \frac{-2 - 3}{2 - (-1)} = \frac{-5}{2 + 1} = \frac{-5}{3}
\]

So, the slope of the line is:

\[
m = -\frac{5}{3}
\]

### Step 2: Use the point-slope form of the line

Now, using the point-slope form \( y - y_1 = m(x - x_1) \), with the point \((-1, 3)\) and the slope \( m = -\frac{5}{3} \), we substitute:

\[
y - 3 = -\frac{5}{3}(x - (-1)) = -\frac{5}{3}(x + 1)
\]

Distribute the slope on the right side:

\[
y - 3 = -\frac{5}{3}x - \frac{5}{3}
\]

Now, add 3 to both sides:

\[
y = -\frac{5}{3}x - \frac{5}{3} + 3
\]

Convert \( 3 \) to a fraction with a denominator of 3:

\[
y = -\frac{5}{3}x - \frac{5}{3} + \frac{9}{3}
\]

Simplify:

\[
y = -\frac{5}{3}x + \frac{4}{3}
\]

### Step 3: The final equation of the line

Thus, the equation of the line is:

\[
y = -\frac{5}{3}x + \frac{4}{3}
\]

### Step 4: Graph verification

The graph provided in the image appears to show a line with a negative slope, and the equation derived matches this expectation.

---

Would you like further clarifications or have specific questions about any step?

### Expanding Questions:
1. How is the slope calculated between two points?
2. Can you explain the point-slope form in more detail?
3. How do you convert the equation from point-slope to slope-intercept form?
4. Why is it important to graph lines when solving linear equations?
5. How would you verify the equation matches the graph?

### Tip:
Always double-check the calculations by substituting the points into the final equation to ensure it satisfies both coordinates.

Derive an equation for the line passing through the points (-1, 3) and (2, -2), and graph the line. Also, write the slope of the line.

Learn how to derive the equation of a line passing through the points (-1, 3) and (2, -2). This problem includes the use of slope formula, point-slope form, and slope-intercept form. Suitable for students in Grades 8-10.

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