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Chapter 7: Problem 54

Graph the plane whose equation is given. $$ 3 x+2 z=9 $$

Short Answer

Expert verified
The plane is parallel to the \(y\)-axis and intersects the \(xz\)-plane at points \((3, 0, 0)\) and \((0, 0, \frac{9}{2})\).

Step by step solution

01

Identify and Rearrange the Equation

The given equation is \(3x + 2z = 9\). Notice there is no \(y\) term, which means that the plane is parallel to the \(y\)-axis. Rearrange the equation to solve for \(z\): \(z = \frac{-3}{2}x + \frac{9}{2}\).
02

Find the x-intercept

To find where the plane intersects the \(x\)-axis, set \(z\) to 0 and solve for \(x\): \(3x + 2(0) = 9\), which simplifies to \(3x = 9\). Solving gives \(x = 3\). The \(x\)-intercept is at the point \((3, 0, 0)\).
03

Find the z-intercept

To find where the plane intersects the \(z\)-axis, set \(x\) to 0 and solve for \(z\): \(3(0) + 2z = 9\), which simplifies to \(2z = 9\). Solving gives \(z = \frac{9}{2}\). The \(z\)-intercept is at the point \((0, 0, \frac{9}{2})\).
04

Sketch the Plane in the xz-plane

Using the intercepts found, plot the points \((3, 0, 0)\) and \((0, 0, \frac{9}{2})\) in the \(xz\)-plane. Draw a line through these points. Since there is no \(y\) dependence, this line extends infinitely parallel to the \(y\)-axis, forming a plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

3D Coordinate System
A 3D coordinate system is essential when dealing with points and objects in three-dimensional space. This system includes three axes: the x-axis, y-axis, and z-axis, which are all perpendicular to each other. Where they intersect is known as the origin, denoted by \((0, 0, 0)\). Each axis allows us to measure distances:
  • x-axis: Represents horizontal movement. Moving right increases the x-value, left decreases it.
  • y-axis: Represents movement into or out of the page or screen. Forward increases the y-value, backward decreases it.
  • z-axis: Represents vertical movement. Upward increases the z-value, downward decreases it.
Understanding this system is crucial when dealing with equations in 3D space, such as the plane described by the equation \(3x + 2z = 9\). By identifying where the plane crosses each axis or lies parallel to an axis, we can graph it correctly.
Intercepts in Mathematics
Intercepts are critical points where a graph intersects an axis. For a 3D coordinate system, you typically look at x, y, and z-intercepts.
  • x-intercept: The point where a graph crosses the x-axis. Here, \(y = 0\) and \(z = 0\).
  • y-intercept: The point where a graph crosses the y-axis. Here, \(x = 0\) and \(z = 0\).
  • z-intercept: The point where a graph crosses the z-axis. Here, \(x = 0\) and \(y = 0\).
In the example equation \(3x + 2z = 9\), we found the intercepts to be at \((3, 0, 0)\) for the x-intercept and \((0, 0, \frac{9}{2})\) for the z-intercept. There's no y-intercept because the plane is parallel to the y-axis, meaning it doesn't touch or cross it. Recognizing intercepts helps plot points necessary for sketching curves or planes in 3D.
Parallel to Axis
When a plane in 3D space is parallel to one of the axes, it does not intersect that axis. This occurs when an axis is absent from the equation as a variable. In our equation \(3x + 2z = 9\), the absence of a \(y\)-term indicates that the plane is parallel to the \(y\)-axis. This means that for any chosen x and z values, the y value can be any number, stretching the plane infinitely in the y-direction. This results in a plane that is not dependent on the y-coordinate, making it flat in the direction of the y-axis.
  • To visualize, take a piece of paper parallel to a wall. The plane (paper) does not touch the wall (y-axis).
  • This allows easy construction of the graph in the remaining two dimensions, x and z.
Understanding axis-parallel planes is essential for clear visualization and plotting in 3D space.

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