To calculate the slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], we use the slope formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Let's consider two points given as examples: [tex]\((1, 2)\)[/tex] and [tex]\((5, 6)\)[/tex].
1. Identify the coordinates:
[tex]\[
(x_1, y_1) = (1, 2)
\][/tex]
[tex]\[
(x_2, y_2) = (5, 6)
\][/tex]
2. Compute the change in [tex]\( y \)[/tex] (rise) and the change in [tex]\( x \)[/tex] (run):
[tex]\[
\text{rise} = y_2 - y_1 = 6 - 2 = 4
\][/tex]
[tex]\[
\text{run} = x_2 - x_1 = 5 - 1 = 4
\][/tex]
3. Substitute these values into the slope formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4}{4} = 1
\][/tex]
So, the slope [tex]\( m \)[/tex] of the line passing through the points [tex]\((1, 2)\)[/tex] and [tex]\((5, 6)\)[/tex] is [tex]\(1\)[/tex].
We now compare this slope to the provided choices:
[tex]\[
m = 4
\][/tex]
[tex]\[
m = \frac{-1}{4}
\][/tex]
[tex]\[
m = \frac{-8}{3}
\][/tex]
[tex]\[
m = \frac{1}{4}
\][/tex]
None of the given choices match our calculated slope of [tex]\(1\)[/tex]. Therefore, the result is [tex]\(1\)[/tex].
