Problem 66 - 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: Recognize the Type of Equation

The equation given is a linear equation in three variables \(x\), \(y\), and \(z\). It is in the form \(Ax + By + Cz = D\), where \(A = 2\), \(B = -3\), \(C = 9\), and \(D = 10\). This form is indicative of a plane in three-dimensional space.

Step 2: Understand the Concept of a Plane

A plane is a flat, two-dimensional surface extending infinitely in three-dimensional space. Every point \((x, y, z)\) that satisfies the equation \(2x - 3y + 9z = 10\) belongs to this plane.

Step 3: Identify Intercepts on Axes

To help visualize this plane, find where it intersects the x, y, and z-axes by setting the other two variables to zero and solving the equation.- For the x-intercept (y = 0, z = 0): \[ 2x = 10 \Rightarrow x = 5 \]- For the y-intercept (x = 0, z = 0): \[ -3y = 10 \Rightarrow y = -rac{10}{3} \]- For the z-intercept (x = 0, y = 0): \[ 9z = 10 \Rightarrow z = rac{10}{9} \]

Step 4: Visualize the Graph

Plot the intercepts found: (5, 0, 0), (0, -\(\frac{10}{3}\), 0), and (0, 0, \(\frac{10}{9}\)). These points lie in the plane described by the equation. Since the equation represents a plane, these points and all the lines connecting them describe the surface of the plane in three-dimensional space.

Step 5: Relate to Real-world Examples

Think of a plane like a sheet of paper extending infinitely, but very thin, within a 3D coordinate system. The tilt and position of the plane are determined by the coefficients of the equation and the constant term (10 in this case).

Key Concepts

Three-Dimensional Graphing

When dealing with linear equations in three variables like \(2x-3y+9z=10\), three-dimensional graphing becomes essential. Unlike two-dimensional equations, these equations describe relationships among three different variables.
To visualize such an equation, we graph it in a three-dimensional coordinate system, with axes labeled \(x\), \(y\), and \(z\). This system is similar to a traditional graph, but it includes all three dimensions:

• The \(x\)-axis represents left and right movement.
• The \(y\)-axis shows forward and backward movement.
• The \(z\)-axis captures up and down motion.

Each point in this system is represented by three coordinates \(x, y, z)\), and these coordinates specify the point's exact location within the space. When you plot multiple points that satisfy a linear equation in three variables, you create a plane in this three-dimensional space.
Graphing in 3D may seem challenging, but the additional dimension helps represent more complex relationships between variables.

Plane in 3D Space

A linear equation in three variables, such as the one given by \(2x - 3y + 9z = 10\), defines a plane in three-dimensional space. A plane is a flat surface that extends infinitely in two directions. It has length and width but no thickness. In the context of a 3D space, it can be imagined as a thin, endless sheet.
Every point \(x, y, z)\) that satisfies the equation belongs to this plane. The equation itself can be rewritten as \(Ax + By + Cz = D\), where \(A\), \(B\), and \(C\) determine the tilt or orientation of the plane, and \(D\) shifts the plane's position within the space.
Planes in 3D space are used extensively in fields like physics, engineering, and computer graphics, wherever spatial reasoning about three dimensions is required. They can model real-life surfaces like walls, floors, and even sheets of paper.

Intercepts on Axes

Finding where a plane intersects the \(x\), \(y\), and \(z\)-axes helps in visualizing its position in space. These points, known as intercepts, are calculated by setting two of the three variables to zero and solving for the remaining variable.

• **For the \(x\)-intercept**: Set \(y = 0\) and \(z = 0\), then solve \(2x = 10\) to find that \(x = 5\). The point is \(5, 0, 0)\).
• **For the \(y\)-intercept**: Set \(x = 0\) and \(z = 0\), then solve \(-3y = 10\) to find that \(y = -\frac{10}{3}\). The point is \(0, -\frac{10}{3}, 0)\).
• **For the \(z\)-intercept**: Set \(x = 0\) and \(y = 0\), then solve \(9z = 10\) to find that \(z = \frac{10}{9}\). The point is \(0, 0, \frac{10}{9})\).

These intercepts illustrate how the plane cuts through each axis, providing valuable reference points. By connecting these intercepts, students can visualize the extent and orientation of the plane in the 3D coordinate system.