To find which line is perpendicular to the line \( y = 5x - 2 \) and goes through the point \( (2, 3) \), we first need to find the slope of the given line.
The slope of the line \( y = 5x - 2 \) is \( 5 \). 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 will be:
\[ -\frac{1}{5} \]
Now, we can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
Where \( m \) is the slope, and \( (x_1, y_1) \) is the point through which the line passes. In this case, \( (x_1, y_1) = (2, 3) \) and \( m = -\frac{1}{5} \).
Substituting these values into the point-slope form:
\[ y - 3 = -\frac{1}{5}(x - 2) \]
Now we can select the corresponding response from the options given:
- \( y + 3 = \frac{1}{5}(x - 2) \)
- \( y - 3 = 5(x - 2) \)
- \( y + 3 = -\frac{1}{5}(x + 2) \)
- \( y - 3 = -\frac{1}{5}(x - 2) \)
Looking at the response options, the correct equation that matches our derived equation \( y - 3 = -\frac{1}{5}(x - 2) \) is:
Response: \( y - 3 = -\frac{1}{5}(x - 2) \).