Problem 65 - Intermediate Algebra

Question

What does the graph of a linear equation in three variables such as \(2 x-3 y+9 z=10\) look like?

Step-by-Step Solution

Answer

The graph is a plane in three-dimensional space.

Step 1: Understanding the Equation

The given equation is a linear equation in three variables: \(2x - 3y + 9z = 10\). This represents a plane in three-dimensional space because a linear equation with three variables describes all points \(x, y, z\) that lie on a flat surface.

Step 2: Determine the Plane's Characteristics

A plane in three dimensions can be described as a flat, two-dimensional surface that extends infinitely in all directions within its dimensions. The equation \(2x - 3y + 9z = 10\) uses coefficients (\(2\), \(-3\), and \(9\)) which serve as the plane's normal vector and influence the orientation and tilt of the plane in 3D space.

Step 3: Visualizing the Plane

To visualize the plane represented by the equation, recognize that it is flat, like a sheet of paper but extended infinitely. The intercepts (where the plane crosses the x, y, or z axis) can be calculated by setting two variables to 0 and solving for the third. However, these intercepts are simply specific points on the plane and do not change its fundamental nature as an infinite flat surface.

Key Concepts

Three-Dimensional Graphing

Three-dimensional graphing involves visualizing mathematical equations in a space with three dimensions: x, y, and z. Unlike the two-dimensional graphs you may be familiar with, 3D graphing requires understanding depth, as an additional z-component is introduced. This allows us to describe objects not just in a flat plane, but in a space where they can have length, width, and height.

In the context of linear equations with three variables like the one presented in our exercise, the graph will manifest as a plane in 3D space. To effectively understand and represent this, one must be comfortable with plotting points that have three coordinates. For instance, the point (2, -3, 5) exists in this 3D world, distinctively defining a position in space.

• If you imagine this space like a real-world room, the corners where the floor meets the walls and ceiling help define your x, y, and z axes.
• Points and lines are plotted in this space to display a broader spectrum of relationships between variables than two-dimensional graphing allows.

Plane in 3D Space

A plane in 3D space is a flat, infinitely extending surface. Any linear equation in three variables, such as the given one in the exercise, represents such a plane. This is because it defines a straight surface that stretches endlessly in all directions within those dimensions.

To imagine a plane, consider a piece of paper. It might be flat and extend only so far physically, yet mathematically, we think of it as stretching out indefinitely. A plane separates the space into parts, contributing to the study of geometry in a spatial context.

• Planes can intersect with each other and lines, creating lines of intersection or single points, but each retains its distinct flat characteristic.
• In equations like the one provided ( 2x - 3y + 9z = 10), the plane it represents contains all points (x, y, z) that satisfy the equation.

Calculating its intercepts with the axes can give you insight into where it crosses the x, y, and z axes, which are useful markers for visualizing the plane's position in space.

Normal Vector of a Plane

The normal vector of a plane plays a crucial role in understanding and graphing planes in 3D space. This vector is essentially a line that is perpendicular to the plane's surface, providing valuable insight into the orientation of the plane itself. In the equation 2x - 3y + 9z = 10, the coefficients 2, -3, and 9 give us the normal vector (2, -3, 9).

Understanding normal vectors is important because:

• They help determine how the plane is tilted in the three-dimensional space.
• A normal vector is orthogonal to every line lying on the plane, making it a key element in understanding the plane's geometry.

To visualize this, consider the normal vector as an arrow pointing directly out from the plane's flat surface. This directional cue helps understand the spatial direction and orientation of the plane, aiding in applications such as engineering and physics, where angles and orientations relative to planes are crucial.