To find the equation of the line that passes through the points $(1,6)$ and $(4,2)$, we can use the point-slope form of a line:

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

Step 1: Find the Slope ($m$)

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

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

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

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

Step 2: Use the Point-Slope Form

Now, plug the slope $m = -\frac{4}{3}$ and one of the points, say $(1,6)$, into the point-slope form:

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

Step 3: Simplify to the Slope-Intercept Form

We can expand and simplify the equation to get it into slope-intercept form $y = mx + b$:

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

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

Since $6$ is $\frac{18}{3}$:

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

Final Equation

The equation of the line in slope-intercept form is:

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

Would you like more details or have any other questions? Here are some related questions:

1. How do you convert a linear equation from point-slope form to standard form?
2. What is the significance of the slope in the equation of a line?
3. How do you determine if two lines are parallel or perpendicular?
4. Can the equation of a line be represented in forms other than slope-intercept and point-slope?
5. How do you find the equation of a line given a point and the slope?

Tip: Always check your slope calculations by substituting the points into the equation to ensure consistency.