To find the equation of the line in point-slope form that passes through the points [tex]\((8,2)\)[/tex] and [tex]\((4,8)\)[/tex], follow these steps:
1. Determine the slope (m):
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, the coordinates of the first point [tex]\((x_1, y_1)\)[/tex] are [tex]\((8,2)\)[/tex], and the coordinates of the second point [tex]\((x_2, y_2)\)[/tex] are [tex]\((4,8)\)[/tex].
Substituting the values:
[tex]\[
m = \frac{8 - 2}{4 - 8} = \frac{6}{-4} = -\frac{3}{2}
\][/tex]
2. Using the slope (m) and point-slope form:
The point-slope form of the equation of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
We can use either of the points [tex]\((8, 2)\)[/tex] or [tex]\((4, 8)\)[/tex] to write our equation.
- Using the point [tex]\((8, 2)\)[/tex]:
[tex]\[
y - 2 = -\frac{3}{2}(x - 8)
\][/tex]
- Using the point [tex]\((4, 8)\)[/tex]:
[tex]\[
y - 8 = -\frac{3}{2}(x - 4)
\][/tex]
Therefore, the equations for the line in point-slope form using the given points are:
[tex]\[
y - 2 = -\frac{3}{2}(x - 8) \quad \text{or} \quad y - 8 = -\frac{3}{2}(x - 4)
\][/tex]
Given the options:
A. [tex]\(y - 2 = -\frac{3}{2}(x - 8)\)[/tex] or [tex]\(y - 8 = -\frac{3}{2}(x - 4)\)[/tex]
B. [tex]\(y + 2 = -\frac{3}{2}(x + 8)\)[/tex] or [tex]\(y + 8 = -\frac{3}{2}(x + 4)\)[/tex]
C. [tex]\(y - 2 = -\frac{3}{2}(x - 4)\)[/tex] or [tex]\(y - 8 = -\frac{3}{2}(x - 8)\)[/tex]
D. [tex]\(y - 2 = 8(x + 8)\)[/tex] or [tex]\(y - 8 = 4(x - 2)\)[/tex]
Option A is the correct one:
[tex]\[
y - 2 = -\frac{3}{2}(x - 8) \quad \text{or} \quad y - 8 = -\frac{3}{2}(x - 4)
\][/tex]
