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Coordinate geometry, sometimes known as analytic geometry, is a mathematical discipline that uses coordinates to describe geometric shapes and properties of space. It is fundamental in connecting algebraic equations to geometric curves and allows for solving geometric problems. In the context of this exercise, coordinate geometry enables us to find the slope of a line.

A coordinate plane is defined by two perpendicular axes, typically labeled as the x-axis (horizontal) and y-axis (vertical). Each point in this plane is represented by a pair of numbers known as coordinates. These coordinates tell us the point's location in respect to the two axes. For example, the point \(1,1\) lies one unit along the x-axis and one unit up the y-axis.

The primary goal in this exercise is to comprehend how these points relate to the line they form and to calculate the slope of that line. This slope is a measure of how steep the line is, providing an overview of how much the y-coordinate changes concerning changes in the x-coordinate.

Linear equations represent straight lines when graphed on a coordinate plane, and they follow the standard form of \(y = mx + c\), where \(m\) is the slope, and \(c\) is the y-intercept. Understanding a line equation means recognizing both directional trend (slope) and vertical shift (y-intercept).

Solving for the slope using two points involves substituting these points into the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].
This formula provides a straightforward method for calculating how much a line "goes up" (or down) for each step it "goes across." As you determine the slope, you effectively figure out the line's rate of change.

In the given exercise, the linear equation involving points \(1, 1\) and \(2, 2\) would suggest a slope (\