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In the world of mathematics, specifically in precalculus and geometry, Cartesian coordinates serve as a cornerstone for plotting points and navigating the plane. They are named after Ren茅 Descartes, who developed the coordinate system. Cartesian coordinates are represented by pairs of numbers \( (x, y) \) that correspond to horizontal and vertical distances on a grid, known as the x-axis and y-axis, respectively.

Imagine a map where every location can be described based on how far east (or west) and north (or south) it is from a starting point. This is essentially what Cartesian coordinates do. They provide a systematic way of describing a point's position in a two-dimensional space. For example, the point \( (-5, 8) \) means it is 5 units to the left and 8 units up from the origin, \( (0, 0) \) where the two axes intersect.

Calculating the distance between two points is much like finding out how far apart two marks are on opposite corners of a rectangle. By using the Cartesian coordinates of these points, we actually have all we need to determine the distance between them using the distance formula. The formula is a direct application of the Pythagorean theorem in a coordinate plane.

The distance formula is expressed as \( d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}} \), where \( x_1 \) and \( y_1 \) are the coordinates of the first point, and \( x_2 \) and \( y_2 \) are the coordinates of the second point. In our exercise, the points \( (-5, 8) \) and \( (-10, 14) \) yield a distance \( d \) which is the length of the hypotenuse of a right-angled triangle with sides \( x_2 - x_1 \) and \( y_2 - y_1 \).

To visualize this, simply draw a horizontal line from each point to drop vertically to the x-axis, and a vertical line to reach over to the y-axis. The rectangle that forms between these lines, as well as the segment connecting the original points, complete the triangle we use for our calculation.

At the heart of calculating distances in coordinate planes lies the Pythagorean theorem, a fundamental principle in geometry. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse \( c \) is equal to the sum of the squares of the lengths of the other two sides \( a \) and \( b \). Formally, it's put as \( c^{2} = a^{2} + b^{2} \).

When applying this theorem to find the distance between two points, think of those points as defining a hypotenuse of a right triangle, while the legs are parallel to the axes. In essence, by using the distance formula, we are using a disguised form of the Pythagorean theorem, where the differences in the x and y coordinates give us the lengths of the sides of the triangle. Through this relationship, we're able to find the straight line distance, which in our exercise example results in \( d = \sqrt{61} \) units after simplification.