Question

select the correct answer. which equation represents the line that is perpendicular to ( y = \frac{1}{5} ) and passes through (-4,-3)? a. ( x = -4 ) b. ( x = -3 ) c. ( y = -5 ) d. ( y = -3 )

Explanation:

Step1: Analyze the slope of the given line

The given line is \( y=\frac{1}{5} \), which is a horizontal line (since it has the form \( y = k \) where \( k=\frac{1}{5} \)). The slope of a horizontal line is \( 0 \). A line perpendicular to a horizontal line is a vertical line. The slope of a vertical line is undefined, and its equation is of the form \( x = c \), where \( c \) is a constant.

Step2: Determine the value of \( c \)

The vertical line passes through the point \( (-4, -3) \). For a vertical line \( x = c \), the \( x \)-coordinate of every point on the line is \( c \). Since the point \( (-4, -3) \) is on the line, the \( x \)-coordinate is \( -4 \), so \( c=-4 \)? Wait, no, wait. Wait, the given line is \( y=\frac{1}{5} \), which is horizontal. A line perpendicular to a horizontal line is vertical, so the equation is \( x = \) the \( x \)-coordinate of the point. Wait, the point is \( (-4, -3) \)? Wait, no, maybe I misread. Wait, the problem says "passes through (-4,-3)"? Wait, no, let's check again. Wait, the original line is \( y=\frac{1}{5} \), which is horizontal (slope 0). A line perpendicular to a horizontal line is vertical, so its equation is \( x = \) constant. The vertical line passes through \( (-4, -3) \), so the \( x \)-coordinate is \( -4 \)? Wait, but the options are A. \( x=-4 \), B. \( x = -3 \), C. \( y=-5 \), D. \( y = -3 \). Wait, no, maybe I made a mistake. Wait, the given line: is it \( y=\frac{1}{5}x \)? Wait, the user wrote "y = 1/5" (maybe a typo, maybe \( y=\frac{1}{5}x \))? Wait, the original problem: "Which equation represents the line that is perpendicular to \( y=\frac{1}{5} \) and passes through (-4,-3)?" Wait, \( y=\frac{1}{5} \) is a horizontal line (slope 0). A line perpendicular to a horizontal line is vertical, so equation \( x = c \). The point is (-4, -3), so \( x=-4 \)? But option A is \( x=-4 \). Wait, but let's confirm. If the given line is \( y=\frac{1}{5} \) (horizontal), then perpendicular is vertical, so \( x = -4 \) (since the point has \( x=-4 \)). So the correct answer is A? Wait, no, wait, maybe the original line is \( y=\frac{1}{5}x \) (slope \( \frac{1}{5} \)). Then the perpendicular slope would be the negative reciprocal, which is \( -5 \). But the options don't have a line with slope -5. So probably the given line is \( y=\frac{1}{5} \) (horizontal). So perpendicular is vertical, equation \( x = -4 \) (since it passes through (-4, -3)). So the answer is A.

Answer:

A. \( x = -4 \)