/*! 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 68 Solve each system of equations u... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 68

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. $$ \left\\{\begin{array}{r} x+2 y-z=3 \\ 2 x-y+2 z=6 \\ x-3 y+3 z=4 \end{array}\right. $$

Short Answer

Expert verified
The system is inconsistent and has no solution.

Step by step solution

01

Write the augmented matrix

Transform the system of equations into an augmented matrix. The system is given by: o (1) \ x + 2y - z = 3 \ (2) \ 2x - y + 2z = 6 \ (3) \ x - 3y + 3z = 4. The augmented matrix is: o \[ \left\[ \begin{array}{ccc|c} 1 & 2 & -1 & 3 \ 2 & -1 & 2 & 6 \ 1 & -3 & 3 & 4 \end{array} \right\]. \]
02

Apply row operations to obtain the row echelon form

Use row operations to simplify the augmented matrix. Start by making the element in \text{row 2, column 1} to zero. Subtract 2 times the first row from the second row:o \[ R2 \rightarrow R2 - 2R1. \]The matrix becomes: o \[ \left\[ \begin{array}{ccc|c} 1 & 2 & -1 & 3 \ 0 & -5 & 4 & 0 \ 1 & -3 & 3 & 4 \end{array} \right\]. \]
03

Eliminate the element in row 3, column 1

Subtract the first row from the third row:o \[ R3 \rightarrow R3 - R1. \]The matrix becomes: o \[ \left\[ \begin{array}{ccc|c} 1 & 2 & -1 & 3 \ 0 & -5 & 4 & 0 \ 0 & -5 & 4 & 1 \end{array} \right\]. \]
04

Eliminate the element in row 3, column 2

Subtract the second row from the third row:o \[ R3 \rightarrow R3 - R2. \]The matrix becomes: o \[ \left\[ \begin{array}{ccc|c} 1 & 2 & -1 & 3 \ 0 & -5 & 4 & 0 \ 0 & 0 & 0 & 1 \end{array} \right\]. \]
05

Determine consistency

The last row of the matrix corresponds to the equation o \[ 0 = 1 \]which is a contradiction. This indicates that the system is inconsistent and has no 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.

Row Operations
Row operations are techniques used to manipulate the rows of a matrix to facilitate solving systems of linear equations. They include three main types:
  • Row swapping: Exchanging two rows
  • Row multiplication: Multiplying all elements of a row by a nonzero scalar
  • Row addition or subtraction: Adding or subtracting a multiple of one row to another row
By strategically applying these operations, we can simplify the matrix to a form that reveals the solutions to the system.
In our exercise, we used row operations to eliminate specific elements to eventually find contradictions, which indicate that the system is inconsistent.
Augmented Matrix
An augmented matrix is a compact representation of a system of linear equations. It includes coefficients of the variables from the left side of each equation, as well as the constants on the right side.
For example, the system of equations:
\( \begin{cases} x + 2y - z = 3 & \ 2x - y + 2z = 6 & \ x - 3y + 3z = 4 \ end{cases} \)
can be written as the augmented matrix:
\[ \begin{array}{ccc|c} 1 & 2 & -1 & 3 \ 2 & -1 & 2 & 6 \ 1 & -3 & 3 & 4 \end{array} \]
This helps to easily apply row operations to solve the system.
Inconsistent System
An inconsistent system of equations is one that has no solution. This occurs when the equations contradict each other, meaning there is no point that satisfies all the equations simultaneously.
In our example, after transforming the augmented matrix through row operations, we arrived at the final form:
\[ \begin{array}{ccc|c} 1 & 2 & -1 & 3 \ 0 & -5 & 4 & 0 \ 0 & 0 & 0 & 1 \end{array} \]
The last row corresponds to the equation \0 = 1, which is clearly a contradiction. This indicates that the system has no possible solutions and is, therefore, inconsistent.
Row Echelon Form
The row echelon form (REF) of a matrix is a type of matrix that is easier to work with for solving systems of equations. It has the following rules:
  • All rows consisting entirely of zeros are at the bottom.
  • In each non-zero row, the leading entry (first non-zero number) is to the right of the leading entry of the row above it.
  • The leading entry in any non-zero row is 1, known as the pivot.
  • The leading entry is the only non-zero entry in its column.
Converting a matrix to REF can greatly simplify solving the system of equations.
In our exercise, through a series of row operations, we aimed to reach such a form to determine the solutions or identify inconsistency.

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

Nutrition A dietitian wishes a patient to have a meal that has 66 grams (g) of protein, 94.5 g of carbohydrates, and 910 milligrams (mg) of calcium. The hospital food service tells the dietitian that the dinner for today is chicken, corn, and \(2 \%\) milk. Each serving of chicken has \(30 \mathrm{~g}\) of protein, \(35 \mathrm{~g}\) of carbohydrates, and \(200 \mathrm{mg}\) of calcium. Each serving of corn has \(3 \mathrm{~g}\) of protein, \(16 \mathrm{~g}\) of carbohydrates, and \(10 \mathrm{mg}\) of calcium. Each glass of \(2 \%\) milk has \(9 \mathrm{~g}\) of protein, \(13 \mathrm{~g}\) of carbohydrates, and \(300 \mathrm{mg}\) of calcium. How many servings of each food should the dietitian provide for the patient?

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{r} 2 x-2 y+3 z=6 \\ 4 x-3 y+2 z=0 \\ -2 x+3 y-7 z=1 \end{array}\right. $$

Computing Wind Speed With a tail wind, a small Piper aircraft can fly 600 miles in 3 hours. Against this same wind, the Piper can fly the same distance in 4 hours. Find the average wind speed and the average airspeed of the Piper.

Determine whether the product is defined. If it is defined, find the product; if it is not write "not defined." $$ \left[\begin{array}{rrr} 4 & -2 & 3 \\ 0 & 1 & 2 \\ -1 & 0 & 1 \end{array}\right]\left[\begin{array}{rr} 2 & 6 \\ 1 & -1 \\ 0 & 2 \end{array}\right] $$

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{r} x-y+z=-4 \\ 2 x-3 y+4 z=-15 \\ 5 x+y-2 z=12 \end{array}\right. $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Decision 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.