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Finding the slope of a line is key to understanding the geometry of straight lines in coordinate geometry. The slope formula is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on the line.
• \( y_2 - y_1 \) represents the change in the \( y \)-coordinates (vertical change).
• \( x_2 - x_1 \) represents the change in the \( x \)-coordinates (horizontal change).

This mathematical relationship shows us how much the line "rises" or "falls" as we move from one point to another along the line. A positive slope means the line is ascending from left to right, while a negative slope indicates that it is descending.

Coordinate geometry, sometimes referred to as analytic geometry, is a way to describe geometry using a coordinate system. It allows us to use algebra to find geometric properties. Points on a plane are identified using ordered pairs, which gives us a way to explore geometric concepts algebraically.In coordinate geometry:

• Each point is defined by coordinates, \((x, y)\), that specify its position within the plane.
• Lines can be illustrated on this plane, and their properties (like slope) can be figured out using algebra.
• The slope formula is a tool from coordinate geometry used to determine how steep a line is.

Performing algebraic calculations is crucial when using the slope formula. When finding a slope, you need to substitute the appropriate values into the formula and carry out careful arithmetic calculations. Let's break it down:1. **Identifying Coordinates**: You first acknowledge which values are \((x_1, y_1)\) and \((x_2, y_2)\). In the given example, \((-2.5, 1.75)\) and \((-0.5, -7.75)\) are your points.2. **Substitution**: Insert these numbers into the slope formula:\[ m = \frac{-7.75 - 1.75}{-0.5 + 2.5} \]This simplifies as you perform the arithmetic operations.3. **Calculate Changes**: Once substituted, calculate the changes:\[ -7.75 - 1.75 = -9.5 \]and \[ -0.5 + 2.5 = 2.0 \]4. **Solve**: Substitute the results back into the slope formula:\[ m = \frac{-9.5}{2.0} \]5. **Final Calculation**: Perform the division to get the slope:\[ m = -4.75 \]This value tells you how much the line decreases by every unit it moves horizontally.Following these steps ensures accuracy, and understanding each calculation step allows for a deep grasp of how lines behave in a coordinate system.