Alright, let's start by identifying the coordinates of the points through which the line passes.
Step 1: Choose [tex]\((x_1, y_1)\)[/tex].
[tex]\[ x_1 = 2, \, y_1 = -5 \][/tex]
Now let's put these values into our table:
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
2 & -5 \\
7 & 1 \\
\hline
\end{tabular}
Next, we will identify the coordinates of the second point.
Step 2: Choose [tex]\((x_2, y_2)\)[/tex].
[tex]\[ x_2 = 7, \, y_2 = 1 \][/tex]
Now, we will calculate the slope of the line.
Step 3: Use the slope formula [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex] where [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are the coordinates of the two points.
Substitute [tex]\(x_1\)[/tex], [tex]\(y_1\)[/tex], [tex]\(x_2\)[/tex], and [tex]\(y_2\)[/tex] into the formula:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
[tex]\[ \text{slope} = \frac{1 - (-5)}{7 - 2} \][/tex]
Step 4: Simplify the numerator and denominator.
[tex]\[
\text{slope} = \frac{1 + 5}{7 - 2} \\
\text{slope} = \frac{6}{5}
\][/tex]
Thus, the slope of the line that passes through the points [tex]\((2, -5)\)[/tex] and [tex]\((7,1)\)[/tex] is [tex]\( \boxed{1.2} \)[/tex].
