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Coordinate geometry is a branch of geometry where we use a coordinate system to define and describe the positions of points, lines, and shapes. In this system, every point is represented by a pair of numbers known as coordinates.

These coordinates describe a point's location on a plane using two values, typically denoted as \(x\) and \(y\). The \(x\) value represents the horizontal position, while the \(y\) value represents the vertical position.

The coordinate plane itself consists of two perpendicular lines called the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin, denoted as (0,0). Using this system makes it easier to analyze geometric figures and perform calculations, such as finding the distance between points or the slope of a line connecting two points.

The slope of a line measures how steep the line is. In coordinate geometry, the slope (often denoted by the letter \(m\)) is calculated using the coordinates of two points on the line. The formula for the slope is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \( (x_1, y_1)\) and \( (x_2, y_2)\) are the coordinates of the two points. The numerator \(y_2 - y_1\) represents the change in the y-coordinates, while the denominator \(x_2 - x_1\) represents the change in the x-coordinates.

If \(m \) is positive, the line slopes upward from left to right. If \(m\) is negative, the line slopes downward from left to right. A larger absolute value of the slope indicates a steeper line. For instance, in our worked example, the slope is calculated as follows:

\[ m = \frac{-3 - (-1)}{5 - 2} = \frac{-3 + 1}{5 - 2} = \frac{-2}{3} \]

Hence, the slope of the line connecting the points (2,-1) and (5,-3) is \ m = -\frac{2}{3} \.

Points on a plane provide a way to visualize and perform calculations with geometric figures in coordinate geometry. To work with points on a plane, follow these steps:

• Identify each point by its coordinates, which are typically in the format (x,y).
• The x-coordinate shows the point's horizontal distance from the y-axis.
• The y-coordinate shows the point's vertical distance from the x-axis.

In our example, the given points are (2,-1) and (5,-3). These coordinates indicate that:

• The point (2,-1) has an x-coordinate of 2 and a y-coordinate of -1.
• The point (5,-3) has an x-coordinate of 5 and a y-coordinate of -3.

By knowing these coordinates, we can perform various calculations like finding the distance between the points or, as in the previous explanation, determining the slope of the line that passes through them. Understanding how to interpret and use points on a plane is essential in coordinate geometry for analyzing and solving geometric problems.