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.