To find a line that is perpendicular to the line \(y = -74x\) and goes through the point \((-7, 5)\), we first need to determine the slope of the given line.
The slope of the line \(y = -74x\) is \(-74\). The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope of the perpendicular line is:
\[ m = -\frac{1}{-74} = \frac{1}{74} \]
Next, we can use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
Here, \((x_1, y_1) = (-7, 5)\) and \(m = \frac{1}{74}\).
Substituting these values into the point-slope formula gives:
\[ y - 5 = \frac{1}{74}(x + 7) \]
Now, let's simplify this equation.
\[ y - 5 = \frac{1}{74}x + \frac{7}{74} \]
Adding 5 to both sides:
\[ y = \frac{1}{74}x + 5 + \frac{7}{74} \]
Next, convert 5 into a fraction with a denominator of 74:
\[ 5 = \frac{370}{74} \]
So,
\[ y = \frac{1}{74}x + \frac{370 + 7}{74} = \frac{1}{74}x + \frac{377}{74} \]
Now, we can find the correct option among the responses provided. However, we see that none of the lines appear to match this final equation.
Let's check to see if perhaps we can determine if one of the provided options has a slope of \( \frac{1}{74} \):
- Option 1: \(y = 47x + 5\) has slope \(47\) (not perpendicular).
- Option 2: \(y = -\frac{7}{4}x + 5\) has slope \(-\frac{7}{4}\) (not perpendicular).
- Option 3: \(y = 47x + 9\) has slope \(47\) (not perpendicular).
- Option 4: \(y = -47x + 1\) has slope \(-47\) (not perpendicular).
None of the given options correspond directly with the slope we calculated. Therefore, it would seem the provided options are incorrect for the condition specified in the question.
In conclusion, the correct line should have a slope of \(\frac{1}{74}\) and pass through the point \((-7, 5)\), which does not match any of the listed responses.