Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the plane represented by the linear equation. Then list four points that lie in the plane. $$x+y+z=8$$

Step-by-Step Solution

Answer

The plane intersects the axes at points (8,0,0), (0,8,0), and (0,0,8). Another point on the plane is (2,3,3). So these are the four points that lie on the plane given by the equation \(x+y+z=8\).

Step 1: Drawing the Plane

First, sketch a 3D coordinate system. Now the equation of a plane is of the form \(Ax+By+Cz=D\), where A, B, C, and D are constants. For the given equation, we have \(x+y+z=8\). This plane intersects the x-axis when y=z=0, the y-axis when x=z=0, and the z-axis when x=y=0. Thus, we have the three points (8,0,0), (0,8,0), and (0,0,8).

Step 2: Locating points on the Plane

Next, choose some points that lie in the plane. For the equation \(x+y+z=8\), any three values that sum up to 8 would satisfy the equation. Let's set: 1. (8,0,0): Here, y and z are 0, and x is 8.2. (0,8,0): Here, x and z are 0, and y is 8.3. (0,0,8): Here, x and y are 0, and z is 8.4. (2,3,3): Here, x is 2, y is 3 and z is 3. All points satisfy the condition \(x+y+z=8\).

Step 3: Verifying the Points

Lastly, cross-check if the selected points meet the equation \(x+y+z=8\). And yes, all four points meet the condition, hence those are correct.

Key Concepts

Equation of a Plane

In 3D coordinate geometry, the **equation of a plane** is typically expressed as \(Ax + By + Cz = D\), where \(A\), \(B\), \(C\), and \(D\) are constants. This form is critical because it represents a flat surface that extends indefinitely in all directions in three-dimensional space.
These constants determine the plane's orientation and position. Given the equation \(x + y + z = 8\), you can observe:

• Each coefficient before the variables is 1, showing equal influence of all three dimensions on the plane.
• The constant 8 shifts the plane away from the origin in space.

The plane can be visualized by finding where it intersects the axes. For instance, when two variables are zero, the third will equal 8.
These intersection points are:

• (8, 0, 0) with the x-axis
• (0, 8, 0) with the y-axis
• (0, 0, 8) with the z-axis

Linear Equations

Linear equations like \(x + y + z = 8\) play a fundamental role in coordinate geometry. These equations imply a linear relationship between the variables.
For a plane, it means that if you move in any straight line direction within the plane, the relationship between changes in \(x\), \(y\), and \(z\) remains constant.

• Such a consistency leads to a flat, two-dimensional surface in three-dimensional space.

The given equation is a straightforward example, illustrating that any combination of \(x\), \(y\), and \(z\) that adds up to 8 lies on the plane.
This consistency allows us to predict or identify points like (2, 3, 3), showing the flexibility of finding points on a linear equation boundary.

3D Sketching

**3D sketching** involves visualizing and drawing objects within a three-dimensional coordinate system. When tackling a problem involving planes, sketching is vital to understand the spatial orientation.
For the equation \(x + y + z = 8\), sketching begins by plotting the basic axes:

• X-axis
• Y-axis
• Z-axis

Each axis provides a guideline for plotting the intersection points identified previously: (8, 0, 0), (0, 8, 0), and (0, 0, 8). Connecting these points in your sketch helps visualize the plane's orientation and position in 3D space.
It's a powerful tool for spatial reasoning, helping you confirm that your mathematical results match their conceptual visualization.

Coordinate Systems

A **coordinate system** in three dimensions is represented by three axes, typically defined as the x-axis, y-axis, and z-axis. These axes are perpendicular to each other, creating a 3D space where any point can be precisely located using a set of three numbers or coordinates.
In our example, the point (2, 3, 3) is found by:

• Moving 2 units along the x-axis,
• 3 units along the y-axis,
• 3 units along the z-axis.

Working with coordinate systems requires understanding that each point's location is relative to the position and orientation of these axes.
It allows you to translate mathematical equations into spatial representations, such as the plane from our original equation. Such visual mapping is crucial in fields like engineering, architecture, and computer graphics, where spatial relationships are essential.