To find the y-intercept of the line passing through the point \((5, -6)\) with a slope of \(-\frac{1}{7}\), we can use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.
Substituting the point \((5, -6)\) and the slope \(-\frac{1}{7}\) into the equation:
\[ y - (-6) = -\frac{1}{7}(x - 5) \]
This simplifies to:
\[ y + 6 = -\frac{1}{7}(x - 5) \]
To isolate \(y\), we can distribute the slope on the right side:
\[ y + 6 = -\frac{1}{7}x + \frac{5}{7} \]
Now, subtract 6 from both sides to solve for \(y\):
\[ y = -\frac{1}{7}x + \frac{5}{7} - 6 \]
We need to express \(-6\) with a denominator of 7, so:
\(-6 = -\frac{42}{7}\)
Thus:
\[ y = -\frac{1}{7}x + \frac{5}{7} - \frac{42}{7} \]
Combine the constants:
\[ y = -\frac{1}{7}x - \frac{37}{7} \]
From the equation \(y = -\frac{1}{7}x - \frac{37}{7}\), we can see that the y-intercept (the value of \(y\) when \(x = 0\)) is:
\[ -\frac{37}{7} \]
Thus, the y-intercept of the line is \(-\frac{37}{7}\).
So the correct answer from the options given is:
-37/7.