Answer :
Let's analyze each statement one by one:
### Statement 1:
The line [tex]\( x=0 \)[/tex] is perpendicular to the line [tex]\( y = -3 \)[/tex].
- [tex]\( x=0 \)[/tex] is a vertical line (it is parallel to the y-axis).
- [tex]\( y=-3 \)[/tex] is a horizontal line (it is parallel to the x-axis).
- Vertical lines and horizontal lines are always perpendicular to each other.
Therefore, Statement 1 is true.
### Statement 2:
All lines that are parallel to the y-axis are vertical lines.
- Lines that are parallel to the y-axis run up and down the plane, which means they’re vertical lines.
Therefore, Statement 2 is true.
### Statement 3:
All lines that are perpendicular to the x-axis have a slope of 0.
- Lines that are perpendicular to the x-axis are vertical lines.
- Vertical lines have an undefined slope, not a slope of 0.
Therefore, Statement 3 is false.
### Statement 4:
The equation of the line parallel to the x-axis that passes through the point [tex]\( (2, -6) \)[/tex] is [tex]\( x=2 \)[/tex].
- A line parallel to the x-axis is a horizontal line.
- A horizontal line passing through the point [tex]\( (2, -6) \)[/tex] has the form [tex]\( y = -6 \)[/tex], not [tex]\( x=2 \)[/tex].
Therefore, Statement 4 is false.
### Statement 5:
The equation of the line perpendicular to the y-axis that passes through the point [tex]\( (-5, 1) \)[/tex] is [tex]\( y=1 \)[/tex].
- A line perpendicular to the y-axis is a horizontal line.
- A horizontal line passing through the point [tex]\( (-5, 1) \)[/tex] has the form [tex]\( y = 1 \)[/tex].
Therefore, Statement 5 is true.
### Conclusion:
The true statements are:
1. The line [tex]\( x=0 \)[/tex] is perpendicular to the line [tex]\( y = -3 \)[/tex].
2. All lines that are parallel to the y-axis are vertical lines.
5. The equation of the line perpendicular to the y-axis that passes through the point [tex]\( (-5, 1) \)[/tex] is [tex]\( y=1 \)[/tex].
Thus, the correct selections are: 1, 2, 5.
### Statement 1:
The line [tex]\( x=0 \)[/tex] is perpendicular to the line [tex]\( y = -3 \)[/tex].
- [tex]\( x=0 \)[/tex] is a vertical line (it is parallel to the y-axis).
- [tex]\( y=-3 \)[/tex] is a horizontal line (it is parallel to the x-axis).
- Vertical lines and horizontal lines are always perpendicular to each other.
Therefore, Statement 1 is true.
### Statement 2:
All lines that are parallel to the y-axis are vertical lines.
- Lines that are parallel to the y-axis run up and down the plane, which means they’re vertical lines.
Therefore, Statement 2 is true.
### Statement 3:
All lines that are perpendicular to the x-axis have a slope of 0.
- Lines that are perpendicular to the x-axis are vertical lines.
- Vertical lines have an undefined slope, not a slope of 0.
Therefore, Statement 3 is false.
### Statement 4:
The equation of the line parallel to the x-axis that passes through the point [tex]\( (2, -6) \)[/tex] is [tex]\( x=2 \)[/tex].
- A line parallel to the x-axis is a horizontal line.
- A horizontal line passing through the point [tex]\( (2, -6) \)[/tex] has the form [tex]\( y = -6 \)[/tex], not [tex]\( x=2 \)[/tex].
Therefore, Statement 4 is false.
### Statement 5:
The equation of the line perpendicular to the y-axis that passes through the point [tex]\( (-5, 1) \)[/tex] is [tex]\( y=1 \)[/tex].
- A line perpendicular to the y-axis is a horizontal line.
- A horizontal line passing through the point [tex]\( (-5, 1) \)[/tex] has the form [tex]\( y = 1 \)[/tex].
Therefore, Statement 5 is true.
### Conclusion:
The true statements are:
1. The line [tex]\( x=0 \)[/tex] is perpendicular to the line [tex]\( y = -3 \)[/tex].
2. All lines that are parallel to the y-axis are vertical lines.
5. The equation of the line perpendicular to the y-axis that passes through the point [tex]\( (-5, 1) \)[/tex] is [tex]\( y=1 \)[/tex].
Thus, the correct selections are: 1, 2, 5.
