To determine the slope of a line that is perpendicular to the line given by the equation [tex]\( y = 6x + 14 \)[/tex], we first need to understand the property of slopes for perpendicular lines.
1. The slope of the given line can be identified directly from its equation in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
In the equation [tex]\( y = 6x + 14 \)[/tex], the slope [tex]\( m \)[/tex] of this line is 6.
2. The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. If the slope of the original line is [tex]\( m \)[/tex], then the slope of a perpendicular line is [tex]\( -\frac{1}{m} \)[/tex].
3. Applying this to our problem, if the slope of the given line [tex]\( y = 6x + 14 \)[/tex] is 6, then the slope of the perpendicular line will be [tex]\( -\frac{1}{6} \)[/tex].
Therefore, the slope of the line that is perpendicular to the line [tex]\( y = 6x + 14 \)[/tex] is [tex]\( -\frac{1}{6} \)[/tex].
So, the correct answer is:
[tex]\[ -\frac{1}{6} \][/tex]
