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Chapter 1: Q6E (page 18)

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss鈥揓ordan elimination. Show all your work.

|x1-7x2+x5=3x3-2x5=2x4+x5=1|

Short Answer

Expert verified

For the given system of equation x5,x5.are free variables and

x3=2+2x5,x4=1-x5,=3+7x2-x5.

Step by step solution

01

Write the augmented matrix

First of all we will make the augmented matrix to the corresponding system of equations.

x1-7x2+x5=3x3-2x5=2x4+x5=1

The corresponding augmented matrix is written as:

1-70010010-200011321

02

Solving the Augmented matrix

The above augmented matrix obtained above is already in reduced form, then the

System of equation to the corresponding above matrix will be

x1=3+7x2-x5x3=2+2x5x4=1-x5

03

Finding free variables

According to system x1=3+7x2-x5x3=2+2x5x4=1-x5, the variables x3,x4,x1depend on x2,x5.So,x2,x5.are free variables.

04

Final answer

Thus, for the given system of equationx2,x5 are free variables and x3=2+2x5,x4=1-x5x1=3+7x2-x5.

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Most popular questions from this chapter

Compute the products Axin Exercises 13 through 15 using

paper and pencil. In each case, compute the product

two ways: in terms of the columns of A and in terms of the rows of A.

14.

[123234][-121]

Consider a solutionx1鈫�of the linear systemAx鈫�=b鈫�. Justify the facts stated in parts (a) and (b):

a. Ifx鈫�his a solution of the systemAx鈫�=0鈫�, thenx1鈫�+xh鈫� is a solution of the systemA=x鈫�=b鈫�.

b. Ifx2鈫�is another solution of the systemAx鈫�=b鈫�, thenx1鈫�+xh鈫�is a solution of the system Ax鈫�+0鈫�.

c. Now suppose A is a2脳2matrix. A solution vectorx1鈫�of the systemAx鈫�+b鈫�is shown in the accompanying figure. We are told that the solutions of the systemAx鈫�=0鈫�form the line shown in the sketch. Draw the line consisting of all solutions of the systemAx鈫�=b鈫�.

If you are puzzled by the generality of this problem, think about an example first:

A=(1鈥呪赌呪赌呪赌�23鈥呪赌呪赌呪赌�6),b鈫�=[39]andx1鈫�=[11]

Students are buying books for the new semester. Brigitte buys the German grammar book and the German novel, Die Leiden des jungen Werther, for袆64 in total. Claude spends袆98 on the linear algebra text and the German grammar book, while Denise buys the linear algebra text and Werther, for 袆76. How much does each of the three books cost?

For an arbitrary positive integern鈮�3, find all solutions x1,x2,x3,......,xnof the simultaneous equations x2=12(x1+x3),x3=12(x2+x4),.....,xn鈭�1=12(xn鈭�2+xn). Note that we are asked to solve the simultaneous equations xk=12(xk鈭�1+xk+1), fork=2,3,.....,n鈭�1 .

Find the least square solutions of the system Ax鈬赌=b鈬赌where

A=[1110-100010-10]andb鈬赌=[110-1010-10].
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