To find the [tex]$y$[/tex]-intercept of a line passing through the point [tex]\((5, -6)\)[/tex] with a slope of [tex]\(-\frac{1}{7}\)[/tex], we will use the slope-intercept form of a linear equation, which is given by:
[tex]\[ y = mx + b \][/tex]
Here, [tex]\(m\)[/tex] is the given slope, and [tex]\(b\)[/tex] is the [tex]$y$[/tex]-intercept that we need to find. The point [tex]\((x_1, y_1)\)[/tex] given to us is [tex]\((5, -6)\)[/tex].
1. Substitute the point [tex]\((5, -6)\)[/tex] and the slope [tex]\(m = -\frac{1}{7}\)[/tex] into the equation to solve for [tex]\(b\)[/tex]:
[tex]\[
-6 = -\frac{1}{7} \cdot 5 + b
\][/tex]
2. First calculate [tex]\(-\frac{1}{7} \cdot 5\)[/tex]:
[tex]\[
-\frac{1}{7} \cdot 5 = -\frac{5}{7}
\][/tex]
3. Now substitute this result into the equation:
[tex]\[
-6 = -\frac{5}{7} + b
\][/tex]
4. To isolate [tex]\(b\)[/tex], add [tex]\(\frac{5}{7}\)[/tex] to both sides of the equation:
[tex]\[
-6 + \frac{5}{7} = b
\][/tex]
5. Convert the integer -6 into a fraction with the same denominator to combine the terms more easily:
[tex]\[
-6 = -\frac{42}{7}
\][/tex]
6. Now, add [tex]\(\frac{5}{7}\)[/tex] to [tex]\(-\frac{42}{7}\)[/tex]:
[tex]\[
-\frac{42}{7} + \frac{5}{7} = \frac{-42 + 5}{7} = \frac{-37}{7}
\][/tex]
So, the [tex]$y$[/tex]-intercept [tex]\(b\)[/tex] is:
[tex]\[
b = -\frac{37}{7}
\][/tex]
Hence, the answer is:
A. [tex]\(-\frac{37}{7}\)[/tex]
