To find the equation of a line that passes through two points, $(x_1, y_1)$ and $(x_2, y_2)$, we can use the slope-intercept form of a line equation:

$y = mx + b$

where $m$ is the slope and $b$ is the y-intercept.

Step 1: Find the slope $m$

The slope $m$ is calculated as:

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

Substituting the points $(2, -3)$ and $(-1, 3)$:

$m = \frac{3 - (-3)}{-1 - 2} = \frac{3 + 3}{-3} = \frac{6}{-3} = -2$

Step 2: Use point-slope form

Now that we have the slope $m = -2$, we can use the point-slope form of a line equation to find the equation:

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

Using the point $(2, -3)$:

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

Simplify the equation:

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

$y + 3 = -2x + 4$

$y = -2x + 4 - 3$

$y = -2x + 1$

Final Equation of the Line:

$y = -2x + 1$

This is the equation of the line that passes through the points $(2, -3)$ and $(-1, 3)$.

Would you like further details or have any questions?

Here are some related questions:

1. How do you convert the equation into standard form?
2. What is the y-intercept of the line?
3. Can you find the equation of a line parallel to this one?
4. How would you determine if another point lies on this line?
5. What is the equation of a line perpendicular to this one through point (2, -3)?

Tip: When finding the equation of a line, always double-check the slope calculation as it can drastically change the result.