Find the slope of a line perpendicular to the line whose equation is 6x + 15y = 225. Fully simplify your answer.
Understand the Problem
The question asks for the slope of a line that is perpendicular to a given line represented by the equation 6x + 15y = 225. To solve this, we will first need to convert the given equation into slope-intercept form (y = mx + b) to identify its slope, and then determine the negative reciprocal of that slope for the perpendicular line.
Answer
The slope of the line perpendicular to the given line is \(m = \frac{5}{2}\).
Answer for screen readers
The slope of the line perpendicular to the given line is (m = \frac{5}{2}).
Steps to Solve
- Convert to Slope-Intercept Form
First, rewrite the equation (6x + 15y = 225) in slope-intercept form (y = mx + b).
To do this, isolate (y):
[ 15y = -6x + 225 ]
Now, divide all terms by 15:
[ y = -\frac{6}{15}x + 15 ]
- Simplify the Slope
Next, simplify the slope (-\frac{6}{15}):
[ -\frac{6}{15} = -\frac{2}{5} ]
So the slope of the given line is (m = -\frac{2}{5}).
- Find the Perpendicular Slope
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, take the negative reciprocal of (-\frac{2}{5}):
The reciprocal of (-\frac{2}{5}) is (-\frac{5}{2}).
Thus, the negative reciprocal is:
[ m_{\perpendicular} = \frac{5}{2} ]
The slope of the line perpendicular to the given line is (m = \frac{5}{2}).
More Information
The concept of perpendicular slopes is based on the fact that the product of the slopes of two perpendicular lines is (-1). This means that if one slope is known, the other can be found by taking its negative reciprocal.
Tips
- Mixing up the process of finding the slope and the perpendicular slope. Always ensure to take the negative reciprocal.
- Forgetting to simplify fractions which can lead to an incorrect answer.