Final answer:
The question revolves around the properties of parallel and perpendicular lines in the cartesian coordinate system. Parallel lines have the same slope, while perpendicular lines intersect at a 90° angle with slopes that are negative reciprocals. The cartesian system can also extend to 3D space with the addition of the z-axis.
Step-by-step explanation:
Parallel and Perpendicular Lines on the Coordinate Plane
Concerning parallel and perpendicular lines on a coordinate plane, it is essential to understand the cartesian coordinate system. Lines that are parallel to the x-axis run horizontally and have the same slope but different y-intercepts. Lines that are perpendicular to each other intersect at a 90° angle, and their slopes are negative reciprocals of each other.
For three lines to be parallel along the x-axis, they must all have the same slope but their y-intercepts can differ. When lines are mutually perpendicular in three-dimensional space, each pair among the x, y, and z axes intersect at right angles. The cartesian system can be extended to three dimensions by introducing the z-coordinate, which represents the vertical axis in three-dimensional space.
Considering the slope of a line, such as in FIGURE A1, if a line has a slope of 3, this means for every 1 unit increase along the x-axis, the line rises by 3 units along the y-axis. The y-intercept is the point where the line crosses the y-axis. In this example, the y-intercept is at 9. Such details are crucial in graphing straight lines and understanding their algebraic properties.