What is the relationship between the \( x \) - and \( y \)-values of each horizontal line segment? Instructions

To understand the relationship between the \( x \)- and \( y \)-values of each horizontal line segment, we can analyze the properties of horizontal lines in a Cartesian coordinate system. 1. **Definition of a Horizontal Line Segment**: A horizontal line segment is a straight line that runs left to right (or right to left) on the graph. It has a constant \( y \)-value for all points along the segment. 2. **Mathematical Representation**: If we denote the horizontal line segment by its endpoints \((x_1, y)\) and \((x_2, y)\), where \(x_1\) and \(x_2\) are the \( x \)-coordinates of the endpoints and \(y\) is the constant \( y \)-coordinate, we can express this relationship as: \[ y = k \] where \(k\) is a constant value representing the \( y \)-coordinate of the line segment. 3. **Relationship Between \( x \) and \( y \)**: - For any point \((x, y)\) on the horizontal line segment, the \( y \)-value remains the same (equal to \( k \)), regardless of the \( x \)-value. - This means that as \( x \) changes (increasing or decreasing), the \( y \)-value does not change. 4. **Conclusion**: The relationship between the \( x \)- and \( y \)-values of each horizontal line segment is that the \( y \)-value is constant while the \( x \)-value can vary. Thus, the horizontal line segment can be described as having the form \( y = k \) for some constant \( k \).