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A coordinate system is a framework for measuring the position of points in a given space. It provides a reference to describe the exact location of something. Imagine a map that shows streets and landmarks. In mathematics, we use a coordinate system to map out points, lines, and shapes on a plane or in space.

For most students, the Cartesian coordinate system is the first encounter they have with plotting points. It includes two perpendicular lines called axes, labeled the x-axis and y-axis in two dimensions. When adding a third dimension, we include the z-axis. These axes meet at a point called the origin, which is the central point of the system and serves as the starting point for measuring distances along the axes.

The x-axis is a horizontal line that represents the first dimension in a coordinate system. It serves as a baseline from which the vertical positions of points are measured.

When plotting a point, the first number in the ordered pair is called the x-coordinate and it shows how far along to the left or right the point is on the x-axis. An x-coordinate with a positive value indicates a position to the right of the origin, while a negative value indicates a position to the left. When the x-coordinate is zero, this means the point lies directly on the y-axis.

Running perpendicular to the x-axis is the y-axis, which is a vertical line representing the second dimension.

The y-coordinate of a point indicates how far up or down it is from the x-axis. A positive y-coordinate means the point is above the x-axis, and a negative y-coordinate means it is below. When the y-coordinate is zero, the point falls on the x-axis itself. Together with the x-axis, the y-axis allows us to plot any point in a two-dimensional plane by using an ordered pair of numbers.

The z-axis comes into play when working within a three-dimensional space. It adds depth to the two-dimensional plane formed by the x-axis and y-axis.

Unlike the x and y-axes which lie on a flat plane, the z-axis can be thought of as coming out towards you or going into the page or screen. The z-coordinate tells us how far in or out a point is relative to this flat plane. In order for a coordinate point to be at the origin where all axes intersect, its z-coordinate must also be zero, just like the x and y coordinates. This third axis is what gives three-dimensional graphics on computers and in games a sense of realism, as they simulate how objects appear from different viewpoints.