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]
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]
