To find a line that is perpendicular to the line \( y = -74x \) and passes through the point \((-7, 5)\), we first need to determine the slope of the given line. The slope of \( y = -74x \) is \(-74\).
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. Thus, the slope of the perpendicular line can be calculated as follows:
\[ \text{slope} = -\frac{1}{-74} = \frac{1}{74} \]
Next, we will use the point-slope form of the line equation, which is given by:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point \((-7, 5)\) and \(m\) is the slope we found (\(\frac{1}{74}\)). Plugging in these values:
\[ y - 5 = \frac{1}{74}(x - (-7)) \]
Simplifying this:
\[ y - 5 = \frac{1}{74}(x + 7) \] \[ y - 5 = \frac{1}{74}x + \frac{7}{74} \] \[ y = \frac{1}{74}x + \frac{7}{74} + 5 \] \[ y = \frac{1}{74}x + \frac{7}{74} + \frac{370}{74} \] \[ y = \frac{1}{74}x + \frac{377}{74} \]
This line in slope-intercept form matches none of the given responses directly, but we can further check which given equations have the slope of \(\frac{1}{74}\).
Now, checking the response options:
- \(y = 47x + 5\) (slope: 47)
- \(y = \frac{4}{7}x + 5\) (slope: \(\frac{4}{7}\))
- \(y = -74x + 5\) (slope: -74)
- \(y = -\frac{7}{4}x + 5\) (slope: -\(\frac{7}{4}\))
- \(y = 47x + 9\) (slope: 47)
- \(y = -47x + 1\) (slope: -47)
None of the provided options appear to directly match our required slope of \(\frac{1}{74}\). It seems that none of those options are correct based on the calculations for a line that is perpendicular to \(y = -74x\) and passes through the point (-7, 5).
Please verify the provided options as they do not reflect the appropriately perpendicular line.