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Coordinates are a set of values that show an exact position. In a two-dimensional plane, coordinates are written as ordered pairs, like \((x, y)\). Here, \(x\) represents the horizontal position and \(y\) represents the vertical position. For example, the coordinates given in the exercise are \((3, -6)\) and \((6, 3)\). These numbers tell you exactly where to place these points on the graph. Understanding coordinates is essential in geometry and algebra to solve various types of problems. By knowing the coordinates, we can easily determine points on a plane, calculate distances, and find midpoints, among other things.

The midpoint of a line segment is the point that divides the segment into two equal lengths. To find it, we use the *midpoint formula*. This formula averages the \((x)\) and \((y)\) coordinates of two endpoints to find the coordinates of the midpoint. The formula is expressed as \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). This means you add the \((x)\) coordinates and \((y)\) coordinates separately, then divide each sum by 2. Using the example from the exercise, we find the midpoint between \((3, -6)\) and \((6, 3)\). By substituting \((x_1 = 3, y_1 = -6, x_2 = 6, y_2 = 3)\) into the formula, you get \(M = \left( \frac{3 + 6}{2}, \frac{-6 + 3}{2} \right)\), which results in \(M = (4.5, -1.5)\).

Algebraic formulas are mathematical equations that use letters and symbols to represent numbers and operations. In the context of midpoints, the midpoint formula is an algebraic formula because it uses variables like \((x_1, x_2, y_1, y_2)\) to generalize the position for any pair of endpoints. Such formulas simplify the calculation process and make it easier to solve complex problems by substituting specific values into a general form. The step-by-step approach to solving the exercise involves identifying the coordinates, applying the formula, and performing the calculations. Understanding these algebraic formulas not only helps in finding midpoints but is also a foundational skill for more advanced mathematical concepts.