/*! 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 39 Use intercepts to help sketch th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 39

Use intercepts to help sketch the plane. \(2 x+5 y+z=10\)

Short Answer

Expert verified
The intercepts are (5, 0, 0), (0, 2, 0), and (0, 0, 10).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \(y\) and \(z\) to zero and solve for \(x\). \(2x + 5(0) + 0 = 10\). Thus, \(2x = 10\) and \(x = 5\). The x-intercept is \((5, 0, 0)\).
02

Find the y-intercept

To find the y-intercept, set \(x\) and \(z\) to zero and solve for \(y\). \(2(0) + 5y + 0 = 10\). Thus, \(5y = 10\) and \(y = 2\). The y-intercept is \((0, 2, 0)\).
03

Find the z-intercept

To find the z-intercept, set \(x\) and \(y\) to zero and solve for \(z\). \(2(0) + 5(0) + z = 10\). Thus, \(z = 10\). The z-intercept is \((0, 0, 10)\).
04

Sketch the plane using intercepts

Plot the intercepts \((5, 0, 0)\), \((0, 2, 0)\), and \((0, 0, 10)\) on a 3D coordinate system. Draw a plane that passes through these three points. This plane represents the equation \(2x + 5y + z = 10\).

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.

Understanding the x-intercept in 3D Geometry
In 3D coordinate geometry, the x-intercept is the point where a plane crosses the x-axis. To find it, we set the other variables, y and z, to zero and solve the equation for x. For the equation \(2x + 5y + z = 10\), we simplify it by substituting zero for y and z, which gives us \(2x = 10\). Solving this equation, we find that \(x = 5\). Therefore, the x-intercept is the point \((5, 0, 0)\).

Key points about the x-intercept:
  • It is found on the x-axis.
  • Other coordinates \(y\) and \(z\) must be zero.
  • Represents the location where the plane first touches or "intersects" the x-axis directly.
Finding the x-intercept helps us understand how a plane is situated in the 3D space and where it meets the x-axis.
Finding the y-intercept in 3D Space
The y-intercept represents the point where the plane intersects the y-axis. To find this point from a given plane equation like \(2x + 5y + z = 10\), set \(x\) and \(z\) to zero. This simplifies our problem to \(5y = 10\). Solving for \(y\), it becomes clear that \(y = 2\). Thus, the y-intercept is \((0, 2, 0)\).

Here is what you need to know about the y-intercept:
  • It exists on the y-axis.
  • With x and z being zero, calculations focus solely on y.
  • It shows the plane's interaction with the y-axis, which is crucial for 3D mapping.
The y-intercept is valuable when sketching planes because it gives us an exact intersection with the y-axis, providing one of the necessary anchors in space.
Discovering the z-intercept
The z-intercept is located where a plane intersects the z-axis. This happens when both x and y are zero. So, for the given equation \(2x + 5y + z = 10\), setting \(x = 0\) and \(y = 0\) simplifies it to \(z = 10\). The z-intercept is then the point \((0, 0, 10)\).

Important characteristics of the z-intercept include:
  • It lies on the z-axis alone.
  • Both x and y values must be zero.
  • The z-intercept defines how the plane reaches out towards the z-direction.
Understanding the z-intercept is important as it helps forge the plane's placement in 3D space, particularly on the z-axis, aiding in accurately sketching the plane's orientation.

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

Find an equation of the plane with \(x\) -intercept \(a, y\) -intercept \(b\) and \(z\) -intercept \(c .\)

Find the area of the parallelogram with vertices \(K(1,2,3),\) \(L(1,3,6), M(3,8,6),\) and \(N(3,7,3)\)

Find direction numbers for the line of intersection of the planes \(x+y+z=1\) and \(x+z=0\)

If \(\mathbf{r}=\langle x, y, z\rangle\) and \(\mathbf{r}_{0}=\left\langle x_{0}, y_{0}, z_{0}\right\rangle,\) describe the set of all points \((x, y, z)\) such that \(\left|\mathbf{r}-\mathbf{r}_{0}\right|=1\)

Find an equation of the plane. The plane through the origin and parallel to the plane \(2 x-y+3 z=1\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.