To find the equation of the line that passes through the points $(-2, 1)$ and $(4, 5)$, we can use the point-slope form of a linear equation. The steps are as follows:

1. Find the Slope ($m$):

The slope $m$ of the line can be calculated using the formula:

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

Substituting the given points $(x_1, y_1) = (-2, 1)$ and $(x_2, y_2) = (4, 5)$:

$m = \frac{5 - 1}{4 - (-2)} = \frac{4}{6} = \frac{2}{3}$

2. Use the Point-Slope Form:

The point-slope form of the equation of a line is:

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

Using the slope $m = \frac{2}{3}$ and one of the points, say $(-2, 1)$:

$y - 1 = \frac{2}{3}(x + 2)$

3. Simplify to Slope-Intercept Form:

Now, we'll simplify this equation to the slope-intercept form $y = mx + b$:

$y - 1 = \frac{2}{3}(x + 2)$ $y - 1 = \frac{2}{3}x + \frac{4}{3}$ $y = \frac{2}{3}x + \frac{4}{3} + 1$ $y = \frac{2}{3}x + \frac{4}{3} + \frac{3}{3}$ $y = \frac{2}{3}x + \frac{7}{3}$

Final Equation:

The equation of the line that passes through the points $(-2, 1)$ and $(4, 5)$ is:

$y = \frac{2}{3}x + \frac{7}{3}$

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the equation of a line passing through the points $(1, 2)$ and $(3, 4)$?
2. How do you find the equation of a line given a point and a slope?
3. What is the slope of the line passing through the points $(0, 0)$ and $(5, 10)$?
4. How can you determine if two lines are parallel using their slopes?
5. What is the slope-intercept form of a line with a slope of $-2$ passing through the point $(1, 3)$?

Tip: When finding the equation of a line from two points, always calculate the slope first, then use one of the points to find the y-intercept.