Select the correct answer.

Which equation represents a line that is parallel to the x-axis, perpendicular to the y-axis, and has a slope of 0?

A. [tex]\( y = \frac{4}{5}x + \frac{5}{4} \)[/tex]
B. [tex]\( y = \frac{5}{4}x \)[/tex]
C. [tex]\( y = \frac{4}{5} \)[/tex]
D. [tex]\( x = \frac{5}{4} \)[/tex]



Answer :

To determine which equation represents a line that is parallel to the [tex]\( x \)[/tex]-axis, perpendicular to the [tex]\( y \)[/tex]-axis, and has a slope of 0, we need to analyze the properties of such a line.

1. Line Parallel to the [tex]\( x \)[/tex]-axis:
- A line that is parallel to the [tex]\( x \)[/tex]-axis has the same [tex]\( y \)[/tex]-coordinate for all values of [tex]\( x \)[/tex]. This means the line is a horizontal line.
- The general form of a horizontal line is [tex]\( y = c \)[/tex], where [tex]\( c \)[/tex] is a constant.

2. Perpendicular to the [tex]\( y \)[/tex]-axis:
- A line that is perpendicular to the [tex]\( y \)[/tex]-axis is the same as being parallel to the [tex]\( x \)[/tex]-axis because it does not change in the [tex]\( y \)[/tex]-direction and extends infinitely in the [tex]\( x \)[/tex]-direction.

3. Slope of 0:
- The slope of a horizontal line is 0. This is because there is no vertical change; the rise over run equals 0.

Let’s analyze the given options:

- Option A: [tex]\( y=\frac{4}{5} x+\frac{5}{4} \)[/tex]
- This is the equation of a line in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] (the slope) is [tex]\( \frac{4}{5} \)[/tex]. Since the slope is not 0, this line is not horizontal.

- Option B: [tex]\( y=\frac{5}{4} x \)[/tex]
- This is also in slope-intercept form [tex]\( y = mx \)[/tex] with a slope [tex]\( m \)[/tex] of [tex]\( \frac{5}{4} \)[/tex]. Again, the slope is not 0, so this line is not horizontal.

- Option C: [tex]\( y=\frac{4}{5} \)[/tex]
- This is in the form [tex]\( y = c \)[/tex] where [tex]\( c \)[/tex] is [tex]\( \frac{4}{5} \)[/tex], a constant. This represents a horizontal line, which means the slope is 0. This line is parallel to the [tex]\( x \)[/tex]-axis and perpendicular to the [tex]\( y \)[/tex]-axis.

- Option D: [tex]\( x=\frac{5}{4} \)[/tex]
- This represents a vertical line where [tex]\( x \)[/tex] is constant. A vertical line is parallel to the [tex]\( y \)[/tex]-axis and does not have a finite slope (its slope is undefined). This does not match the conditions given.

Based on these analyses, the correct equation representing a line parallel to the [tex]\( x \)[/tex]-axis, perpendicular to the [tex]\( y \)[/tex]-axis, and with a slope of 0 is:

Option C: [tex]\( y=\frac{4}{5} \)[/tex]