To find the equation of the line that is perpendicular to [tex]\( y = -\frac{1}{6} x + 4 \)[/tex] and passes through the point [tex]\((-1, 4)\)[/tex], let's go through the solution step-by-step:
1. Determine the slope of the given line:
The given line's equation is [tex]\( y = -\frac{1}{6} x + 4 \)[/tex]. The slope ([tex]\( m \)[/tex]) of this line is [tex]\( -\frac{1}{6} \)[/tex].
2. Find the slope of the perpendicular line:
Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of [tex]\( -\frac{1}{6} \)[/tex] is:
[tex]\[
\text{slope of perpendicular line} = -\frac{1}{\left(-\frac{1}{6}\right)} = 6
\][/tex]
3. Use the point-slope form of the equation:
The point-slope form of a line's equation is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point on the line. Here, [tex]\( m = 6 \)[/tex] and the point [tex]\((x_1, y_1)\)[/tex] is [tex]\((-1, 4)\)[/tex].
Plug these values into the point-slope form:
[tex]\[
y - 4 = 6(x - (-1))
\][/tex]
Simplify the equation:
[tex]\[
y - 4 = 6(x + 1)
\][/tex]
[tex]\[
y - 4 = 6x + 6
\][/tex]
4. Solve for [tex]\( y \)[/tex] to write the equation in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = 6x + 6 + 4
\][/tex]
[tex]\[
y = 6x + 10
\][/tex]
Therefore, the equation of the line that is perpendicular to [tex]\( y = -\frac{1}{6} x + 4 \)[/tex] and passes through the point [tex]\((-1, 4)\)[/tex] is:
[tex]\[
y = 6x + 10
\][/tex]
So we can write this plainly as:
[tex]\[
y = 6x + 10
\][/tex]
