Find the Line Equation Through Points (2,8) and (6,1): Coordinate Geometry

To find the equation of the line passing through the points $(2,8)$ and $(6,1)$, follow the steps below:

• Step 1: Calculate the slope $m$.
  The formula for the slope $m$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 8}{6 - 2} = \frac{-7}{4}$

• Step 2: Use the point-slope form to write the equation of the line.
  The point-slope form of a line is given by:

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

Using the point $(2,8)$:

$y - 8 = -\frac{7}{4}(x - 2)$

• Step 3: Simplify to the slope-intercept form.
  Distribute the slope and rearrange to find $y$:

$y - 8 = -\frac{7}{4}x + \frac{7}{2}$

Add 8 to both sides to solve for $y$:

$y = -\frac{7}{4}x + \frac{7}{2} + 8$

Convert 8 to a fraction with a denominator of 2:

$y = -\frac{7}{4}x + \frac{7}{2} + \frac{16}{2}$

Simplify the addition:

$y = -\frac{7}{4}x + \frac{23}{2}$

To convert $23/2$ into mixed number form: $\frac{23}{2} = 11 \frac{1}{2}$

Thus, the equation in slope-intercept form is: $y = -\frac{7}{4}x + 11 \frac{1}{2}$

Therefore, the equation of the line passing through these points is $y = -1\frac{3}{4}x + 11\frac{1}{2}$, which matches the correct choice in the multiple-choice answers.