/*! 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 5 Find the intercepts and sketch t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 5

Find the intercepts and sketch the graph of the plane. $$ 2 x-y+3 z=4 $$

Short Answer

Expert verified
The x-intercept is 2, the y-intercept is -4, and the z-intercept is \(4/3\). So, the plane passes through the points (2, 0, 0), (0, -4, 0), and (0, 0, 4/3).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set the other variables (y and z) equal to zero in the equation \(2x - y + 3z = 4\). This simplifies to \(2x = 4\). Solving for x evaluates to \(x = 2\).
02

Find the y-intercept

Likewise, the y-intercept is found by setting x and z equal to zero. The equation becomes \(-y = 4\), thereby yielding \(y = -4\).
03

Find the z-intercept

Similarly, to find the z-intercept, both x and y are set equal to zero. This simplifies the equation to \(3z = 4\), which solves to \(z = 4/3\).
04

Sketch the graph

After obtaining all the intercepts (2, -4, 4/3), plot these points on a three-dimensional coordinate system. The plane passing through these intercept points is the required solution.

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
The x-intercept of a plane in 3D coordinate geometry is where the plane intersects the x-axis. To find the x-intercept, we usually set the other two variables, y and z, to zero and solve for x in the given plane equation.
This removes their influence, making it simple to isolate x.For example, considering the plane equation from our problem: \(2x - y + 3z = 4\), you replace y and z with zero, getting:\(2x = 4\).
This simplifies to \(x = 2\).
So, when the plane crosses the x-axis, it will be at the point (2, 0, 0).
  • Set y = 0 and z = 0.
  • Reduce the equation to x terms.
  • Solve to find the value of x; this gives the x-intercept.
y-intercept
The y-intercept in 3D geometry indicates where a plane intersects the y-axis. To determine this, we set x and z to zero. This approach isolates y in the equation, showcasing its placement along the y-axis.Using the given equation: \(2x - y + 3z = 4\), we plug in x = 0 and z = 0, transforming it to:\(-y = 4\).This simplifies to \(y = -4\).
Hence, the y-intercept is at the point (0, -4, 0) on the y-axis.
  • Set x and z to zero.
  • Simplify to focus on y.
  • Solve to find the y-intercept.
z-intercept
The z-intercept for a plane in three-dimensional coordinates is located where the plane touches the z-axis. To identify this, x and y are each set to zero in the plane equation, highlighting z's role.Taking the equation \(2x - y + 3z = 4\), inputting x = 0 and y = 0 reduces it to:\(3z = 4\).
Solving this gives \(z = \frac{4}{3}\).
Thus, the plane meets the z-axis at (0, 0, \(\frac{4}{3}\)).
  • Assign zero to both x and y.
  • Simplify to emphasize z-terms.
  • Determine the z-intercept through solving.
graph of a plane
Sketching the graph of a plane in 3D geometry involves plotting its intercepts and visualizing the surface extending through these points. This process emphasizes the plane's spatial positioning within a coordinate system.With the points (2, 0, 0), (0, -4, 0), and (0, 0, \(\frac{4}{3}\)), the task is to mark these on a three-dimensional graph properly.Understanding Graph Drawing:
  • Start by plotting each intercept point.
  • Visualize a flat surface connecting these points.
  • This represents a single slice of the infinite plane.
Practically, this drawing step helps in grasping how planes extend beyond their intercepts in all dimensions. It鈥檚 like mapping a flat sheet that runs endlessly through the intercepts you've identified.

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

Use a double integral to find the volume of the solid bounded by the graphs of the equations. $$ z=x^{2}, z=0, x=0, x=2, y=0, y=4 $$

Examine the function for relative extrema and saddle points. $$ f(x, y)=x^{2}-y^{2}+4 x-4 y-8 $$

Set up the integral for both orders of integration and use the more convenient order to evaluate the integral over the region \(R .\) \(\int_{R} \int \frac{y}{x^{2}+y^{2}} d A\) \(R:\) triangle bounded by \(y=x, y=2 x, x=2\)

Sketch the region of integration and evaluate the double integral. $$ \int_{0}^{a} \int_{0}^{\sqrt{a^{2}-x^{2}}} d y d x $$

Examine the function for relative extrema and saddle points. $$ f(x, y)=3 x^{2}+2 y^{2}-12 x-4 y+7 $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.