To find the equation of the line passing through the point $(-2, 2)$ and the midpoint of the line segment joining the points $(4, 3)$ and $(-2, 5)$, we can follow these steps:

Step 1: Find the Midpoint of the Line Segment

The midpoint $(M)$ of a line segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Substituting the given points $(4, 3)$ and $(-2, 5)$:

$M = \left(\frac{4 + (-2)}{2}, \frac{3 + 5}{2}\right) = \left(\frac{2}{2}, \frac{8}{2}\right) = (1, 4)$

Step 2: Find the Slope of the Line

The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

Using the points $(-2, 2)$ and $(1, 4)$:

$m = \frac{4 - 2}{1 - (-2)} = \frac{2}{3}$

Step 3: Use the Point-Slope Form to Find the Equation

The point-slope form of a line is:

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

Using the slope $m = \frac{2}{3}$ and the point $(-2, 2)$:

$y - 2 = \frac{2}{3}(x - (-2))$

Simplify this equation:

$y - 2 = \frac{2}{3}(x + 2)$

$y - 2 = \frac{2}{3}x + \frac{4}{3}$

$y = \frac{2}{3}x + \frac{4}{3} + 2$

$y = \frac{2}{3}x + \frac{4}{3} + \frac{6}{3}$

$y = \frac{2}{3}x + \frac{10}{3}$

Final Equation of the Line

$\boxed{y = \frac{2}{3}x + \frac{10}{3}}$

Would you like further details or have any questions?

Related Questions:

1. How to find the equation of a line given two points directly?
2. What is the significance of the midpoint in coordinate geometry?
3. How to convert a point-slope form to slope-intercept form?
4. How does changing one point affect the slope of the line?
5. What are other forms of linear equations besides slope-intercept and point-slope?

Tip: Always double-check the calculated midpoint and slope before forming the final equation to avoid simple calculation errors!