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When graphing in three dimensions, we use a 3D coordinate system. This system extends the familiar 2D coordinate plane (which has x and y axes) by adding a third axis, the z-axis. In this space, every point is represented by three coordinates:

• The x-coordinate measures the point's horizontal distance from the origin (0,0,0).
• The y-coordinate measures the vertical distance from the x-z plane.
• The z-coordinate measures the distance above or below the x-y plane.

The origin in 3D is still the center where all three axes intersect. To visualize this, imagine the z-axis shooting up into the sky from the flat surface of the 2D plane. This addition allows us to explore geometry and equations in a broader sense, such as surfaces and planes rather than just lines and shapes that exist in two dimensions. Understanding how to navigate and plot within this 3D space is crucial for visualizing and solving equations involving planes.

In three-dimensional graphing, the x-intercept is the point where the graph of an equation crosses the x-axis. To clarify, the x-axis is the horizontal line in a 3D space where both y and z are 0. To determine the x-intercept:

• Set y = 0: This means the point lies in the x-z plane.
• Set z = 0: This means the point exactly lies on the x-axis.

Substitute these values into your equation to find the x-coordinate. For instance, using the equation we have, when y = 0 and z = 0, the equation becomes:\[3x + 0 = 6\]This simplifies to x = 2, giving us the x-intercept at the point (2, 0, 0). This point is significant as it helps define the location and orientation of a plane in conjunction with other intercepts.

The z-intercept of an equation in a 3D coordinate system is the point where the graph intersects the z-axis. The z-axis runs perpendicular to both the x and y axes, extending from the plane they create. To find the z-intercept, both x and y are set to 0:

• Set x = 0: This means moving along the line parallel to the z-axis.
• Set y = 0: This keeps us on the x-z plane.

Substitute these into the equation to solve for the value of z. From the example given, when both x and y are 0, the equation simplifies:\[0x + 0 + z = 6\]Thus, z = 6, giving us the z-intercept at the point (0, 0, 6). This point is not only a part of defining the shape of the plane in a 3D space but also essential in plotting and understanding the plane's dimensions and location relative to the origin.