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Coordinate geometry, often known as analytic geometry, bridges algebra and geometry by using a coordinate system. This approach allows one to solve geometric problems through algebraic equations. The basic building blocks of coordinate geometry are points. A point on a two-dimensional plane has two coordinates: an 'x' coordinate representing the horizontal position, and a 'y' coordinate representing the vertical position.

In the example provided, the line is defined by parametric equations which represent points on the line as functions of a parameter 't'. By selecting different values of 't', we calculate distinct points on the line, thereby mapping it onto the plane. This demonstrates how coordinate geometry can be used to represent lines algebraically and then translate them into a visual format.

Algebraic calculations form the foundation of manipulating mathematical expressions to find solutions. These calculations can involve operations like addition, subtraction, multiplication, and division, along with the use of variables and constants. The essence of algebra is finding unknown values through the properties of equality and arithmetic operations.

In the context of the exercise, algebraic calculations are used to find the coordinates of points on a line expressed in parametric form. For instance, by substituting chosen values for the parameter 't' into the parametric equations, one performs operations to solve for 'x' and 'y'. Such direct calculations show the practical application of algebra in finding specific solutions to geometric problems.

Graphing lines is a vital concept in both algebra and geometry, providing visual representation of solutions and helping to understand the behavior of linear functions. A line can be graphed using various methods; however, with parametric equations, it involves calculating coordinates at specific values of the parameter and plotting them.

In the exercise given, after calculating the points (2,1) for when 't' is 0, and (5,-1) for when 't' is 1, plotting these points on a coordinates plane provides a visual representation of the line. Connecting the dots, we get the line's graph, showing how the values of 't' lead to the movement along the line. Graphing is not only about the end-result, but also understanding the relationship between the algebraic formulas and the geometric figures they represent.