To find the equation of the line that passes through the points $(-3, 5)$, $(x, y)$, and $(-2, 3)$, follow these steps:

1. Calculate the slope between two points

First, calculate the slope ($m$) of the line using the points $(-3, 5)$ and $(-2, 3)$. The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the coordinates $(x_1, y_1) = (-3, 5)$ and $(x_2, y_2) = (-2, 3)$ into the formula:

$m = \frac{3 - 5}{-2 - (-3)} = \frac{-2}{-2} = 1$

So, the slope $m$ is $1$.

2. Use the slope-point form to find the equation of the line

Now that we have the slope $m = 1$, we can use the point-slope form of the equation of a line, which is:

$y - y_1 = m(x - x_1)$

Substitute the slope $m = 1$ and the coordinates of one of the points, say $(-3, 5)$, into the equation:

$y - 5 = 1(x - (-3))$

Simplify the equation:

$y - 5 = x + 3$

$y = x + 8$

3. Verify the equation with the second point

We found the equation $y = x + 8$. To verify, substitute the second point $(-2, 3)$ into this equation:

$3 = -2 + 8$

$3 = 6$

This result seems incorrect, meaning an error occurred. Let's re-evaluate the slope calculation and equation.

4. Re-calculate the correct slope and line equation

The previous steps revealed a possible miscalculation, so let's find the slope again between $(-3, 5)$ and $(-2, 3)$:

$m = \frac{3 - 5}{-2 - (-3)} = \frac{-2}{1} = -2$

Now, using point-slope form with this corrected slope:

$y - 5 = -2(x + 3)$

Simplify the equation:

$y - 5 = -2x - 6$

$y = -2x - 1$

5. Verify with the second point

Substituting $(-2, 3)$ into $y = -2x - 1$:

$3 = -2(-2) - 1 = 4 - 1 = 3$

This matches, confirming that the correct equation of the line is:

$\boxed{y = -2x - 1}$

Would you like further details or have any other questions?

Here are 5 related questions to explore further:

1. How do you determine if three points lie on the same straight line?
2. What is the general form of the equation of a line?
3. How do you find the equation of a line parallel to a given line through a specific point?
4. What is the relationship between the slopes of perpendicular lines?
5. How can you derive the equation of a line from its slope and a point?

Tip: Always double-check your slope calculations and use a consistent approach to verify the correctness of your linear equations.