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Coordinate geometry, often referred to as analytic geometry, melds algebra and geometry to discuss the positions and shapes of figures. It's like painting a picture on a grid using numbers. Here, each point on the plane is described as a pair of numbers known as coordinates. These coordinates are written as \(x, y\), representing horizontal and vertical positions respectively on the \(x\)-axis and \(y\)-axis.

In exercises involving the slope of a line, coordinate geometry helps us visualize. For instance, given the points (-3,5) and (9,5), they both have the same second coordinate (5) on the y-axis. This tells us both points lie on a horizontal line parallel to the x-axis! To find any line's characteristics, like its slope, we simply need the coordinates of two points on it.

Linear equations model relationships as straight lines on a graph. In algebra, these are easily recognizable in the form \(y = mx + b\), where \m\ denotes the slope, and \b\ represents the y-intercept, where the line crosses the y-axis.

To determine the slope of such lines, especially when given two points, use the formula for slope: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This formula essentially measures the change in 'y' versus the change in 'x' between two points. A slope quantifies the line's tilt, showing how much y increases as x increases. If the slope equals zero, like in our exercise, it means the line is horizontal, indicating no vertical change as x changes.

Mathematical formulas are crucial tools that provide shortcuts to complex calculations. In finding the slope, the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) is a go-to:

• The numerator \(y_2 - y_1\) signifies the difference in y-values of the two given points, essentially the line's vertical movement.
• The denominator \(x_2 - x_1\) captures the difference between the x-values, which indicates the horizontal movement.

By inserting the coordinates \((-3, 5)\) and \(9, 5)\) from our exercise into this formula, the numerator becomes zero (since \(5 - 5 = 0\)), leading to the slope of zero. Therefore, in scenarios where y-values remain consistent across different x-values, you are dealing with a horizontal line, embodying a slope of zero. Formulas simplify these derivations, pulling together our understanding of motion and position into neat calculations.