/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 Label any intercepts and sketch ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 63

Label any intercepts and sketch a graph of the plane.\(4 x+2 y+6 z=12\)

Short Answer

Expert verified
The intercepts of the equation \(4x+2y+6z=12\) are: X-intercept at (3,0,0), Y-intercept at (0,6,0), and Z-intercept at (0,0,2). After plotting these points, one can see they form a triangle in the 3-dimensional coordinate system, which represents the plane.

Step by step solution

01

Find the X-intercept

To find the X-intercept, one can set y and z to 0 in the equation. Thus, the equation becomes \(4x+2(0)+6(0)=12\), and solving this gives \(x=3\). Therefore, the X-intercept is at the point (3,0,0).
02

Find the Y-intercept

To find the Y-intercept, x and z are set to 0 in the equation. This gives \(4(0)+2y+6(0)=12\), which simplifies into \(2y=12\), so \(y=6\). Thus, the Y-intercept is (0,6,0).
03

Find the Z-intercept

For finding Z-intercept, x and y are set to 0. Putting these values in the equation gives \(4(0)+2(0)+6z=12\), which leads to \(z=2\). Therefore, the Z-intercept is (0,0,2).
04

Sketch the Graph

Beginning with a 3-dimensional coordinate system, one can then plot the X, Y, and Z intercepts which are at points of (3,0,0), (0,6,0), and (0,0,2) respectively. Once these points are plotted, connecting them will form a triangle that represents the given plane in 3 dimensional space.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

X-intercept
To find the X-intercept of a plane in a 3D coordinate system, we start by setting the other variables to zero. In the equation of the plane, which is given as \( 4x + 2y + 6z = 12 \), the X-intercept occurs when both \( y \) and \( z \) are zero. By substituting these values into the equation, it simplifies to \( 4x = 12 \).
This straightforward calculation gives us \( x = 3 \), identifying the X-intercept as the point \( (3, 0, 0) \).
Understanding the concept of the X-intercept involves:
  • Setting the other variables (\( y \) and \( z \)) to zero.
  • Solving the equation for \( x \).
  • Visualizing this point where the plane intersects the x-axis.
This point records where the plane cuts through the x-axis in three-dimensional space. It鈥檚 crucial for graphing and comprehending the spatial orientation of the plane.
Y-intercept
The Y-intercept is found by setting the \( x \) and \( z \) variables to zero in the plane's equation. For our equation \( 4x + 2y + 6z = 12 \), doing this transforms the equation into \( 2y = 12 \). Solving this results in \( y = 6 \), indicating that the Y-intercept is at the point \( (0, 6, 0) \).
Locating this intercept involves a straightforward method:
  • Substituting zero for \( x \) and \( z \).
  • Solving for \( y \).
  • Plotting this point on the y-axis.
This point illustrates where the plane intersects the y-axis. Visualizing the location of the Y-intercept helps in connecting the dots needed to sketch the plane within the 3D coordinate grid.
Z-intercept
Finding the Z-intercept consists of setting both \( x \) and \( y \) to zero in the plane's equation. Taking the equation \( 4x + 2y + 6z = 12 \), it becomes \( 6z = 12 \) with these substitutions. Solving it for \( z \) gives \( z = 2 \), marking the Z-intercept as \( (0, 0, 2) \).
Here are the steps of identifying the Z-intercept:
  • Setting \( x \) and \( y \) values to zero in the equation.
  • Solving for \( z \).
  • Identifying and plotting this point on the z-axis.
The Z-intercept marks where the plane intersects the z-axis. Visualizing this intercept is an important step in understanding how the plane interacts with each of the coordinate system axes. This understanding allows for an accurate sketch of the plane's orientation in the 3D space.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Write an equation whose graph consists of the set of points \(P(x, y, z)\) that are twice as far from \(A(0,-1,1)\) as from \(B(1,2,0)\)

Find the angle between the diagonal of a cube and the diagonal of one of its sides.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Identify the curve of intersection of the surfaces (in spherical coordinates) \(\rho=2 \sec \phi\) and \(\rho=4\).

Sketch the solid that has the given description in spherical coordinates. \(0 \leq \theta \leq 2 \pi, \pi / 4 \leq \phi \leq \pi / 2,0 \leq \rho \leq 1\)

Give the standard equation of a plane in space. Describe what is required to find this equation.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.