To find the line perpendicular to [tex]\( y = -\frac{1}{3}x + 7 \)[/tex] that passes through the point [tex]\( (4, 2) \)[/tex], follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\( y = -\frac{1}{3}x + 7 \)[/tex]. The coefficient of [tex]\( x \)[/tex], which is [tex]\( -\frac{1}{3} \)[/tex], is the slope of this line.
2. Determine the slope of the perpendicular line:
Two lines are perpendicular if and only if the product of their slopes is [tex]\( -1 \)[/tex]. Therefore, the slope [tex]\( m \)[/tex] of the line perpendicular to [tex]\( y = -\frac{1}{3}x + 7 \)[/tex] must satisfy:
[tex]\[
m \cdot \left( -\frac{1}{3} \right) = -1
\][/tex]
Solve for [tex]\( m \)[/tex]:
[tex]\[
m = -1 \div \left( -\frac{1}{3} \right) = 3
\][/tex]
3. Use the point-slope form of the line equation:
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope. In this case, the slope [tex]\( m \)[/tex] is [tex]\( 3 \)[/tex] and the point [tex]\( (x_1, y_1) \)[/tex] is [tex]\( (4, 2) \)[/tex].
4. Substitute the slope and the given point into the point-slope form:
Substitute [tex]\( m = 3 \)[/tex], [tex]\( x_1 = 4 \)[/tex], and [tex]\( y_1 = 2 \)[/tex] into the point-slope form equation:
[tex]\[
y - 2 = 3(x - 4)
\][/tex]
Therefore, the equation of the line perpendicular to [tex]\( y = -\frac{1}{3}x + 7 \)[/tex] that passes through the point [tex]\( (4, 2) \)[/tex] is:
[tex]\[
y - 2 = 3(x - 4)
\][/tex]
