Alright, let's write the equation of the line that passes through the points [tex]\((7, 6)\)[/tex] and [tex]\((-1, 2)\)[/tex]. To do this, we will follow these steps:
1. Determine the slope [tex]\(m\)[/tex] of the line:
The formula for the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the points [tex]\((7, 6)\)[/tex] and [tex]\((-1, 2)\)[/tex]:
[tex]\[
m = \frac{2 - 6}{-1 - 7} = \frac{-4}{-8} = \frac{1}{2}
\][/tex]
2. Identify the y-intercept [tex]\(b\)[/tex] of the line:
The slope-intercept form of a line is given by [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept. We can use one of the points and the slope to solve for [tex]\(b\)[/tex]. Let's use the point [tex]\((7, 6)\)[/tex]. Substitute [tex]\(m = \frac{1}{2}\)[/tex], [tex]\(x = 7\)[/tex] and [tex]\(y = 6\)[/tex] into the equation:
[tex]\[
6 = \frac{1}{2}(7) + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
6 = \frac{7}{2} + b
\][/tex]
[tex]\[
6 - \frac{7}{2} = b
\][/tex]
[tex]\[
\frac{12}{2} - \frac{7}{2} = b
\][/tex]
[tex]\[
\frac{5}{2} = b
\][/tex]
3. Write the equation in slope-intercept form:
Now that we have the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(b\)[/tex], we can write the equation of the line:
[tex]\[
y = \frac{1}{2}x + \frac{5}{2}
\][/tex]
So, the equation of the line that passes through [tex]\((7, 6)\)[/tex] and [tex]\((-1, 2)\)[/tex] in slope-intercept form is:
[tex]\[
\boxed{y = \frac{1}{2} x + \frac{5}{2}}
\][/tex]
Among the given options, the correct one is:
[tex]\[
y = \frac{1}{2} x + \frac{5}{2}
\][/tex]
