The given line equation is [tex]\( y = -\frac{1}{3}x - \frac{1}{3} \)[/tex].
1. Determine the slope of the given line:
- The equation is in slope-intercept form ([tex]\( y = mx + b \)[/tex]), where [tex]\( m \)[/tex] is the slope.
- Here, the slope [tex]\( m \)[/tex] of the given line is [tex]\( -\frac{1}{3} \)[/tex].
2. Find the slope of the perpendicular line:
- The slope of a line perpendicular to another line is the negative reciprocal of the given line's slope.
- The negative reciprocal of [tex]\( -\frac{1}{3} \)[/tex] is [tex]\( 3 \)[/tex].
3. Use the point-slope form to find the equation of the perpendicular line passing through the point [tex]\( (2, -1) \)[/tex]:
- Point-slope form 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.
- Here, [tex]\( m = 3 \)[/tex], [tex]\( x_1 = 2 \)[/tex], and [tex]\( y_1 = -1 \)[/tex].
4. Substitute the values into the point-slope form:
[tex]\( y - (-1) = 3(x - 2) \)[/tex]
[tex]\( y + 1 = 3(x - 2) \)[/tex]
5. Solve for [tex]\( y \)[/tex] to get the equation in slope-intercept form ([tex]\( y = mx + b \)[/tex]):
[tex]\( y + 1 = 3x - 6 \)[/tex]
[tex]\( y = 3x - 6 - 1 \)[/tex]
[tex]\( y = 3x - 7 \)[/tex]
Therefore, the equation of the line that is perpendicular to the given line and passes through the point [tex]\( (2, -1) \)[/tex] is [tex]\( y = 3x - 7 \)[/tex].
The correct answer is the fourth option:
[tex]\[ y = 3x - 7 \][/tex]
