To find the equation of a line that is perpendicular to the line given by the equation \(y = -3x + 5\), we first need to determine the slope of the original line. The slope of the line \(y = -3x + 5\) is \(-3\).
Lines that are perpendicular to each other have slopes that are negative reciprocals. The negative reciprocal of \(-3\) is \(\frac{1}{3}\).
Now we can examine the provided options to find which equation has a slope of \(\frac{1}{3}\):
- \(y = 3x + 9\) (slope = 3)
- \(y = -\frac{1}{3}x + 9\) (slope = -\frac{1}{3})
- \(y = -3x + 9\) (slope = -3)
- \(y = \frac{1}{3}x + 9\) (slope = \frac{1}{3})
The correct choice is:
y = \frac{1}{3}x + 9
This line has the slope of \(\frac{1}{3}\), which is perpendicular to the slope of \(-3\).