To find the equation of the line passing through the points $(1, 6)$ and $(4, 2)$, follow these steps:

Step 1: Find the slope (m) of the line

The slope $m$ is given by:

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

Substituting the given points $(x_1, y_1) = (1, 6)$ and $(x_2, y_2) = (4, 2)$:

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

Step 2: Use the point-slope form of the line equation

The point-slope form of a line's equation is:

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

Substituting $m = -\frac{4}{3}$ and using the point $(1, 6)$:

$y - 6 = -\frac{4}{3}(x - 1)$

Step 3: Simplify the equation

Distribute the slope on the right side:

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

Now, add 6 to both sides to get the equation in slope-intercept form $y = mx + b$:

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

Combine the constants on the right side:

$y = -\frac{4}{3}x + \frac{4}{3} + \frac{18}{3} = -\frac{4}{3}x + \frac{22}{3}$

Final Equation

The equation of the line is:

$y = -\frac{4}{3}x + \frac{22}{3}$

Would you like a more detailed explanation or have any questions?

Here are 5 related questions you might find interesting:

1. How do you find the equation of a line given one point and the slope?
2. What is the importance of the slope-intercept form of a line?
3. How do you convert a linear equation from point-slope form to standard form?
4. How can you verify if a given point lies on a line using its equation?
5. What happens to the slope if the line passes through two points with the same x-coordinates?

Tip: When working with slopes, remember that a negative slope indicates the line is decreasing (downward) as it moves from left to right.