Question

Label any intercepts and sketch a graph of the plane. $$ 2 x-y+z=4 $$

Short answer

The x-intercept is at (2,0,0), the y-intercept is at (0,-4,0), and the z-intercept is at (0,0,4). The plane sketched using these intercepts is a triangle in three-dimensional space.

Step-by-step solution

Finding the x-intercept

This can be found by setting \(y = 0\) and \(z = 0\) in the equation, which will leave only \(x\). On solving the equation \(2x - 0 + 0 = 4\), the value of \(x = 2\) is obtained.

Finding the y-intercept

This can be found by setting \(x = 0\) and \(z = 0\) in the equation, which will leave only \(y\). On solving the equation \(2(0) - y + 0 = 4\), the value of \(y = -4\) is obtained.

Finding the z-intercept

This can be found by setting \(x = 0\) and \(y = 0\) in the equation, which will leave only \(z\). On solving the equation \(2(0) - 0 + z = 4\), the value of \(z = 4\) is obtained.

Sketch the plane

Now that we have the intercepts, we can sketch the plane. The x-intercept is at (2,0,0), the y-intercept is at (0,-4,0), and the z-intercept is at (0,0,4). These points are plotted in the three-dimensional space and joined to form a triangular plane.

Key concepts

X-Intercept

Understanding the concept of an x-intercept is crucial when learning about graphing in three dimensions. The x-intercept is the point where a line or plane crosses the x-axis in a 3D coordinate system. To find this point, we hold the other variables constant—typically setting them to zero—and solve for x. Let's consider the equation of a plane 2x - y + z = 4. To find the x-intercept, set y and z to zero, resulting in the equation 2x = 4. Solving for x gives us x = 2, which means the x-intercept is at the point (2, 0, 0).

This point is one of the three we need to graph the plane. It represents where the plane touches the x-axis, and pictorially, it's a spot on the axis. It's helpful to imagine sliding along the x-axis from the origin until you reach this intercept.

Y-Intercept

The y-intercept is where a line or plane intersects the y-axis, and similarly to finding the x-intercept, we find the y-intercept by considering the equation 2x - y + z = 4 and setting the other variables to zero. In this case, setting x and z to zero and solving for y gives us -y = 4 or y = -4.

The y-intercept is therefore located at (0, -4, 0). This point, along with the x and z intercepts, helps us define the position and orientation of a plane in a 3D space. In practical terms, the y-intercept shows us where the plane would hit if we were moving along the y-axis, starting from the origin.

Z-Intercept

Similarly, the z-intercept is the point where the plane crosses the z-axis. Using our same equation, we set the other variables, in this case x and y, to zero. The solution of the equation z = 4 reveals that the z-intercept is at the point (0, 0, 4).

This point completes the trio of intercepts and is essential for creating a visual of the plane in three dimensions. When the concept of a z-intercept is understood, students grasp how to move along the z-axis from the origin and visualize the point in space where the plane would intersect it.

3D Coordinate System

A 3D coordinate system is a framework for determining the position of points or objects in three-dimensional space. This system uses three axes—commonly labeled x, y, and z—that intersect at the origin (0, 0, 0) and stretch out infinitely in positive and negative directions.

Each axis represents a dimension: the x-axis typically runs horizontally, the y-axis vertically, and the z-axis denotes depth. Every point in this 3D system is identified by an ordered triplet (x, y, z), where each value corresponds to the position on the respective axis. For instance, the intercepts we found, (2, 0, 0), (0, -4, 0), and (0, 0, 4), tell us the exact spots where the plane crosses the corresponding axes in this 3-dimensional setup.

By plotting these intercepts and understanding the system they lie in, students are better equipped to sketch the plane and grasp the nature of objects in a 3D environment.