Certainly! Let’s work through the problem step-by-step.
### Part I: Calculating the Slope
To determine the slope [tex]\( m \)[/tex] of the line passing through the points [tex]\((6, 5)\)[/tex] and [tex]\((3, 1)\)[/tex], we use the slope formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, [tex]\((x_1, y_1) = (6, 5)\)[/tex] and [tex]\((x_2, y_2) = (3, 1)\)[/tex].
Substituting these values into the slope formula:
[tex]\[
m = \frac{1 - 5}{3 - 6} = \frac{-4}{-3} = \frac{4}{3}
\][/tex]
So, the slope of the line is:
[tex]\[
m = \frac{4}{3}
\][/tex]
### Part II: Writing Two Point-Slope Form Equations
The point-slope form of the equation of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
We will write two equations using the given points [tex]\((6, 5)\)[/tex] and [tex]\((3, 1)\)[/tex].
#### Using the point [tex]\((6, 5)\)[/tex]:
Substitute [tex]\( x_1 = 6 \)[/tex], [tex]\( y_1 = 5 \)[/tex], and [tex]\( m = \frac{4}{3} \)[/tex] into the point-slope form:
[tex]\[
y - 5 = \frac{4}{3}(x - 6)
\][/tex]
#### Using the point [tex]\((3, 1)\)[/tex]:
Substitute [tex]\( x_2 = 3 \)[/tex], [tex]\( y_2 = 1 \)[/tex], and [tex]\( m = \frac{4}{3} \)[/tex] into the point-slope form:
[tex]\[
y - 1 = \frac{4}{3}(x - 3)
\][/tex]
### Summary:
1. Slope: [tex]\( m = \frac{4}{3} \)[/tex]
2. Point-slope equation using point [tex]\((6, 5)\)[/tex]:
[tex]\[
y - 5 = \frac{4}{3}(x - 6)
\][/tex]
3. Point-slope equation using point [tex]\((3, 1)\)[/tex]:
[tex]\[
y - 1 = \frac{4}{3}(x - 3)
\][/tex]
These are the detailed steps to calculate the slope and to write the two point-slope form equations for the line passing through the given points.
