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A horizontal line on a graph means that the line runs left to right without any slant. All points along a horizontal line have the same y-coordinate. In other words, no matter how much you change the x-value, the y-value stays constant.
For example, in our given exercise, the horizontal line passes through the point \((5,1)\). This means every point on this line has a y-coordinate of 1. The x-coordinate can be any value, but it won't affect the y-coordinate. Understanding this concept is important for grasping other topics in coordinate geometry.

The slope of a line measures how steep the line is. It is found using the slope formula: \( m = \frac{\Delta y}{\Delta x} \). Here, \(m\) represents the slope, \(\Delta y\) (delta y) is the change in the y-coordinates (rise), and \(\Delta x\) (delta x) is the change in the x-coordinates (run).
For a horizontal line, there is no vertical change, which means \(\Delta y = 0\). Therefore, the slope formula becomes \( m = \frac{0}{\Delta x} = 0 \). No vertical change means the slope is 0. In our example, no matter what values the x-coordinates may take, the y-coordinate does not change. This is why the slope of the horizontal line passing through \((5,1)\) is 0.

Coordinate geometry, also known as analytic geometry, helps us study geometry using a coordinate system. In this system, each point on the plane is described using an ordered pair of numbers (x, y).
The x-coordinate tells us how far to move left or right from the origin (0,0), and the y-coordinate tells us how far to move up or down. In the context of our exercise, the point \((5,1)\) lies 5 units to the right of the origin and 1 unit up. The concept of slope is central to coordinate geometry as it helps describe how lines tilt or slant in the plane.
By understanding these fundamental concepts, students can solve more complex geometric problems and better understand the relationships between different geometric figures in coordinate geometry.