Question

The points (6,5) and (8,2) fall on a particular line. What is its equation in point-slope form?

Answer 1

To find the equation of the line in point-slope form, we first need to calculate the slope \( m \) of the line connecting the two points \( (6, 5) \) and \( (8, 2) \).

The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the coordinates of the points:

- \( (x_1, y_1) = (6, 5) \)
- \( (x_2, y_2) = (8, 2) \)

We calculate:

\[
m = \frac{2 - 5}{8 - 6} = \frac{-3}{2}
\]

Now that we have the slope \( m = -\frac{3}{2} \), we can use one of the points to write the equation in point-slope form. We'll use the point \( (6, 5) \).

The point-slope form of a line is given by:

\[
y - y_1 = m(x - x_1)
\]

Substituting \( m = -\frac{3}{2} \), \( x_1 = 6 \), and \( y_1 = 5 \):

\[
y - 5 = -\frac{3}{2}(x - 6)
\]

So the equation of the line in point-slope form is:

\[
y - 5 = -\frac{3}{2}(x - 6)
\]