Answer :
To determine which equation describes a line that passes through the point [tex]\((1,2)\)[/tex] and is parallel to the [tex]\(x\)[/tex]-axis, we need to understand the characteristics of such a line.
1. Understanding a Line Parallel to the [tex]\(x\)[/tex]-axis:
- A line that is parallel to the [tex]\(x\)[/tex]-axis does not change its [tex]\(y\)[/tex]-value regardless of the [tex]\(x\)[/tex]-coordinate.
- This means the line will have a constant [tex]\(y\)[/tex]-value for all points on the line.
2. Identifying the Equation:
- Given the point [tex]\((1,2)\)[/tex], the [tex]\(y\)[/tex]-value of the line must be 2.
- Therefore, the equation of the line must be [tex]\(y = 2\)[/tex], as this equation ensures that all points on the line will have the same [tex]\(y\)[/tex]-value of 2, ensuring it is parallel to the [tex]\(x\)[/tex]-axis.
3. Reviewing the Options:
- Option A: [tex]\(y = 2\)[/tex] — This satisfies our requirement since it maintains a constant [tex]\(y\)[/tex]-value of 2.
- Option B: [tex]\(y = 1\)[/tex] — This would represent a line parallel to the [tex]\(x\)[/tex]-axis but passing through [tex]\(y = 1\)[/tex], not the point [tex]\((1,2)\)[/tex].
- Option C: [tex]\(x = 2\)[/tex] — This represents a vertical line parallel to the [tex]\(y\)[/tex]-axis, not parallel to the [tex]\(x\)[/tex]-axis.
- Option D: [tex]\(x = 1\)[/tex] — This also represents a vertical line parallel to the [tex]\(y\)[/tex]-axis, not parallel to the [tex]\(x\)[/tex]-axis.
Thus, the correct equation of the line passing through the point [tex]\((1,2)\)[/tex] and parallel to the [tex]\(x\)[/tex]-axis is:
A. [tex]\(y=2\)[/tex]
1. Understanding a Line Parallel to the [tex]\(x\)[/tex]-axis:
- A line that is parallel to the [tex]\(x\)[/tex]-axis does not change its [tex]\(y\)[/tex]-value regardless of the [tex]\(x\)[/tex]-coordinate.
- This means the line will have a constant [tex]\(y\)[/tex]-value for all points on the line.
2. Identifying the Equation:
- Given the point [tex]\((1,2)\)[/tex], the [tex]\(y\)[/tex]-value of the line must be 2.
- Therefore, the equation of the line must be [tex]\(y = 2\)[/tex], as this equation ensures that all points on the line will have the same [tex]\(y\)[/tex]-value of 2, ensuring it is parallel to the [tex]\(x\)[/tex]-axis.
3. Reviewing the Options:
- Option A: [tex]\(y = 2\)[/tex] — This satisfies our requirement since it maintains a constant [tex]\(y\)[/tex]-value of 2.
- Option B: [tex]\(y = 1\)[/tex] — This would represent a line parallel to the [tex]\(x\)[/tex]-axis but passing through [tex]\(y = 1\)[/tex], not the point [tex]\((1,2)\)[/tex].
- Option C: [tex]\(x = 2\)[/tex] — This represents a vertical line parallel to the [tex]\(y\)[/tex]-axis, not parallel to the [tex]\(x\)[/tex]-axis.
- Option D: [tex]\(x = 1\)[/tex] — This also represents a vertical line parallel to the [tex]\(y\)[/tex]-axis, not parallel to the [tex]\(x\)[/tex]-axis.
Thus, the correct equation of the line passing through the point [tex]\((1,2)\)[/tex] and parallel to the [tex]\(x\)[/tex]-axis is:
A. [tex]\(y=2\)[/tex]
