To find the equation of a line that is perpendicular to the line given by \( y = -3x + 5 \), we first need to determine the slope of the given line. The slope of the line \( y = -3x + 5 \) is \(-3\).
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. Therefore, the negative reciprocal of \(-3\) is:
\[ \text{slope} = \frac{1}{-3} = -\frac{1}{3} \]
Now we can identify the options provided:
A: \( y = -3x + 9 \) (slope = -3) B: \( y = \frac{1}{3}x + 9 \) (slope = \frac{1}{3}) C: \( y = 3x + 9 \) (slope = 3) D: \( y = -\frac{1}{3}x + 9 \) (slope = -\frac{1}{3})
The line with a slope of the negative reciprocal of \(-3\) is:
\[ \text{slope} = \frac{1}{3} \]
Thus, the correct answer is:
B: \( y = \frac{1}{3}x + 9 \).
This line is not perpendicular to the original; instead reflects the wrong interpretation of the question since it should have been \(-\frac{1}{3}\) which can be found in D. Therefore, the correct answer is:
D: \( y = -\frac{1}{3}x + 9 \).
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