/*! 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} Q10E In Exercises 10, write a vector ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Q10E (page 1)

In Exercises 10, write a vector equation that is equivalent tothe given system of equations.

10. \(4{x_1} + {x_2} + 3{x_3} = 9\)

\(\begin{array}{c}{x_1} - 7{x_2} - 2{x_3} = 2\\8{x_1} + 6{x_2} - 5{x_3} = 15\end{array}\)

Short Answer

Expert verified

The system of equations in the vector equation form is\({x_1}\left[ {\begin{array}{*{20}{c}}4\\1\\8\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}1\\{ - 7}\\6\end{array}} \right] + {x_3}\left[ {\begin{array}{*{20}{c}}3\\{ - 2}\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}9\\2\\{15}\end{array}} \right]\).

Step by step solution

01

Write the system of equations in a single vector

Consider the system of equations as shown below:

\(\begin{array}{c}4{x_1} + {x_2} + 3{x_3} = 9\\{x_1} - 7{x_2} - 2{x_3} = 2\\8{x_1} + 6{x_2} - 5{x_3} = 15\end{array}\)

Write the left-hand part and the right-hand part of the system of equations in the vector form as shown below:

\(\left[ \begin{array}{c}4{x_1} + {x_2} + 3{x_3}\\{x_1} - 7{x_2} - 2{x_3}\\8{x_1} + 6{x_2} - 5{x_3}\end{array} \right] = \left[ {\begin{array}{*{20}{c}}9\\2\\{15}\end{array}} \right]\)

02

Separate the left-hand side of the system of equations

The left-hand side of the vector equation contains \(\left[ \begin{array}{c}4{x_1} + {x_2} + 3{x_3}\\{x_1} - 7{x_2} - 2{x_3}\\8{x_1} + 6{x_2} - 5{x_3}\end{array} \right]\). Write it as sums of vectors in terms of unknowns \({x_1}\), \({x_2}\), and \({x_3}\).

\(\left[ {\begin{array}{*{20}{c}}{4{x_1}}\\{{x_1}}\\{8{x_1}}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{{x_2}}\\{ - 7{x_2}}\\{6{x_2}}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{3{x_3}}\\{ - 2{x_3}}\\{ - 5{x_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}9\\2\\{15}\end{array}} \right]\)

03

Take the unknowns’ common from each vector

It is observed that \({x_1}\) is the common in the vector \(\left[ {\begin{array}{*{20}{c}}{4{x_1}}\\{{x_1}}\\{8{x_1}}\end{array}} \right]\). Take \({x_1}\) as the common from the first vector as shown below:

\({x_1}\left[ {\begin{array}{*{20}{c}}4\\1\\8\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{{x_2}}\\{ - 7{x_2}}\\{6{x_2}}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{3{x_3}}\\{ - 2{x_3}}\\{ - 5{x_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}9\\2\\{15}\end{array}} \right]\)

04

Take the unknowns’ common from each vector

Similarly, take \({x_2}\) as the common from the vector \(\left[ {\begin{array}{*{20}{c}}{{x_2}}\\{ - 7{x_2}}\\{6{x_2}}\end{array}} \right]\), and \({x_3}\) as the common from the vector \(\left[ {\begin{array}{*{20}{c}}{3{x_3}}\\{ - 2{x_3}}\\{ - 5{x_3}}\end{array}} \right]\). Take \({x_2}\) and \({x_3}\) as commons from each vector as shown below:

\({x_1}\left[ {\begin{array}{*{20}{c}}4\\1\\8\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}1\\{ - 7}\\6\end{array}} \right] + {x_3}\left[ {\begin{array}{*{20}{c}}3\\{ - 2}\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}9\\2\\{15}\end{array}} \right]\)

Thus, the system of equations in the vector equation form is:

\({x_1}\left[ {\begin{array}{*{20}{c}}4\\1\\8\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}1\\{ - 7}\\6\end{array}} \right] + {x_3}\left[ {\begin{array}{*{20}{c}}3\\{ - 2}\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}9\\2\\{15}\end{array}} \right]\)

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!

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

Construct a \(3 \times 3\) matrix\(A\), with nonzero entries, and a vector \(b\) in \({\mathbb{R}^3}\) such that \(b\) is not in the set spanned by the columns of\(A\).

Consider the problem of determining whether the following system of equations is consistent:

\(\begin{aligned}{c}{\bf{4}}{x_1} - {\bf{2}}{x_2} + {\bf{7}}{x_3} = - {\bf{5}}\\{\bf{8}}{x_1} - {\bf{3}}{x_2} + {\bf{10}}{x_3} = - {\bf{3}}\end{aligned}\)

  1. Define appropriate vectors, and restate the problem in terms of linear combinations. Then solve that problem.
  1. Define an appropriate matrix, and restate the problem using the phrase 鈥渃olumns of A.鈥�
  1. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.

An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let \({T_1},...,{T_4}\) denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodes鈥攖o the left, above, to the right, and below. For instance,

\({T_1} = \left( {10 + 20 + {T_2} + {T_4}} \right)/4\), or \(4{T_1} - {T_2} - {T_4} = 30\)

33. Write a system of four equations whose solution gives estimates

for the temperatures \({T_1},...,{T_4}\).

In Exercises 3 and 4, display the following vectors using arrows

on an \(xy\)-graph: u, v, \( - {\bf{v}}\), \( - 2{\bf{v}}\), u + v , u - v, and u - 2v. Notice that u - vis the vertex of a parallelogram whose other vertices are u, 0, and \( - {\bf{v}}\).

4. u and v as in Exercise 2

Show that if ABis invertible, so is B.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics 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.