To find the equation of a line that is perpendicular to a given line, you need to understand the relationship between the slopes of the two lines. The slope of the line you provided, \( y = -6x + 2 \), is \( -6 \).
For two lines to be perpendicular, the product of their slopes must equal -1. Therefore, if the slope of the given line is \( m_1 = -6 \), the slope \( m_2 \) of the perpendicular line can be calculated using the formula:
\[ m_1 \cdot m_2 = -1 \]
Substituting the known slope:
\[ -6 \cdot m_2 = -1 \]
To find \( m_2 \):
\[ m_2 = \frac{-1}{-6} = \frac{1}{6} \]
Now that we have the slope of the perpendicular line, we can express its equation in point-slope form, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. You can choose any point for the perpendicular line. For example, let's choose the point \( (0, 0) \) (the origin).
Substituting the slope \( m_2 = \frac{1}{6} \) and the point \( (0, 0) \):
\[ y - 0 = \frac{1}{6}(x - 0) \]
This simplifies to:
\[ y = \frac{1}{6}x \]
Thus, one equation of a line that is perpendicular to \( y = -6x + 2 \) is:
\[ y = \frac{1}{6}x \]
You can also express this in various forms or choose other points, but the slope will remain \( \frac{1}{6} \).