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Coordinate geometry, also known as analytic geometry, is an area of mathematics that describes the position of points, lines, and shapes through a coordinate system. A common system used is the Cartesian coordinate system where any point in a plane is identified by unique x (horizontal) and y (vertical) coordinates.

In the case of finding the slope of a line, coordinate geometry allows us to quantify how steep a line is in the plane. To do so, we consider two points on the line, for instance, (1, 2) and (3, 3) in our exercise. These points have specific x and y coordinates that we use in the slope formula to calculate the steepness of the line. This process provides a numerical measure of the incline (or decline) of a line which is fundamental in various applications such as physics, engineering, and computer graphics.

The slope of a line in coordinate geometry is given by the slope formula, which is expressed as\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. This formula calculates the rate of change in the y-coordinate relative to the x-coordinate between two points on a line. Simply put, it tells us how much the y-value increases or decreases as the x-value increases by one.

Using the formula, when you substitute the coordinates of our two points (1, 2) for \(x_1, y_1\) and (3, 3) for \(x_2, y_2\), you apply it as:\[ m = \frac{3 - 2}{3 - 1} = \frac{1}{2} \]. This calculation tells us that for every one unit increase in x, y increases by one-half unit, exemplifying a relatively gentle slope. The slope is positive, indicating the line rises as one moves from left to right.

Linear equations form the basis for lines in coordinate geometry. These equations can be written in various forms, such as the point-slope form, the slope-intercept form, and the standard form. The most common representation of a linear equation is the slope-intercept form, which is given by \[ y = mx + b \], where \(m\) is the slope of the line and \(b\) is the y-intercept, the point at which the line crosses the y-axis.

The slope we calculated using the slope formula (\(m = \frac{1}{2}\)) can be used as the 'm' in the slope-intercept form of a linear equation. If we know the y-intercept or another point on the line, we can easily write the equation of the line. Understanding how to work with the slope and different forms of linear equations is essential in solving a variety of problems related to lines in the coordinate plane.