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 \(m = \frac{5}{2}\).

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.

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