Step 1

Since you have asked multiple question, we will solve the first question for you. If you want any specific question to be solved then please specify the question number or post only that question.

Consider the following system of linear equations:

2x−3y+7z=5

3x+y−3z=13

2x+19y−47z=32

Rewrite the system of equations in augmented matrix form:

\(\displaystyle{\left[\begin{array}{ccc|c} {2}&-{3}&{7}&{5}\\{3}&{1}&-{3}&{13}\\{2}&{19}&-{47}&{32}\end{array}\right]}\)

Step 2

\(\displaystyle{D}={\left|\begin{array}{ccc} {2}&-{3}&{7}\\{3}&{1}&-{3}\\{2}&{19}&-{47}\end{array}\right|}\)

\(\displaystyle={2}{\left|\begin{array}{cc} {1}&-{3}\\{19}&-{47}\end{array}\right|}-{\left(-{3}\right)}{\left|\begin{array}{cc} {3}&-{3}\\{2}&-{47}\end{array}\right|}+{7}{\left|\begin{array}{cc} {3}&{1}\\{2}&{19}\end{array}\right|}\)

\(\displaystyle={2}{\left[{1}\times{\left(-{47}\right)}-{\left(-{3}\right)}\times{19}\right]}+{3}{\left[{3}\times{\left(-{47}\right)}-{\left(-{3}\right)}\times{2}\right]}+{7}{\left({3}\times{19}-{1}\times{2}\right)}\)

=2(-47+57)+3(-141+6)+7(57-2)

\(\displaystyle={2}\times{10}+{3}\times{\left(-{135}\right)}+{7}\times{55}\)

D=20 - 405 + 385

=0

Step 3

\(\displaystyle{D}_{{x}}={\left|\begin{array}{ccc} {5}&-{3}&{7}\\{13}&{1}&-{3}\\{32}&{19}&-{47}\end{array}\right|}\)

\(\displaystyle={5}{\left|\begin{array}{cc} {1}&-{3}\\{19}&-{47}\end{array}\right|}-{\left(-{3}\right)}{\left|\begin{array}{cc} {13}&-{3}\\{32}&-{47}\end{array}\right|}+{7}{\left|\begin{array}{cc} {13}&{1}\\{32}&{19}\end{array}\right|}\)

\(\displaystyle={5}{\left[{1}\times{\left(-{47}\right)}-{\left(-{3}\right)}\times{19}\right]}+{3}{\left[{13}\times{\left(-{47}\right)}-{\left(-{3}\right)}\times{32}\right]}+{7}{\left({13}\times{19}-{1}\times{32}\right)}\)

=5(-47+57)+3(-611+96)+7(247-32)

\(\displaystyle={5}\times{10}+{3}\times{\left(-{515}\right)}+{7}\times{215}\)

\(\displaystyle{D}_{{x}}={50}-{1545}+{1505}\)

=10

D=0 and \(\displaystyle{D}_{{x}}\ne{0}\).

Hence, the system of equations is inconsistent.

Hence, the system of equations has no solution.

Since you have asked multiple question, we will solve the first question for you. If you want any specific question to be solved then please specify the question number or post only that question.

Consider the following system of linear equations:

2x−3y+7z=5

3x+y−3z=13

2x+19y−47z=32

Rewrite the system of equations in augmented matrix form:

\(\displaystyle{\left[\begin{array}{ccc|c} {2}&-{3}&{7}&{5}\\{3}&{1}&-{3}&{13}\\{2}&{19}&-{47}&{32}\end{array}\right]}\)

Step 2

\(\displaystyle{D}={\left|\begin{array}{ccc} {2}&-{3}&{7}\\{3}&{1}&-{3}\\{2}&{19}&-{47}\end{array}\right|}\)

\(\displaystyle={2}{\left|\begin{array}{cc} {1}&-{3}\\{19}&-{47}\end{array}\right|}-{\left(-{3}\right)}{\left|\begin{array}{cc} {3}&-{3}\\{2}&-{47}\end{array}\right|}+{7}{\left|\begin{array}{cc} {3}&{1}\\{2}&{19}\end{array}\right|}\)

\(\displaystyle={2}{\left[{1}\times{\left(-{47}\right)}-{\left(-{3}\right)}\times{19}\right]}+{3}{\left[{3}\times{\left(-{47}\right)}-{\left(-{3}\right)}\times{2}\right]}+{7}{\left({3}\times{19}-{1}\times{2}\right)}\)

=2(-47+57)+3(-141+6)+7(57-2)

\(\displaystyle={2}\times{10}+{3}\times{\left(-{135}\right)}+{7}\times{55}\)

D=20 - 405 + 385

=0

Step 3

\(\displaystyle{D}_{{x}}={\left|\begin{array}{ccc} {5}&-{3}&{7}\\{13}&{1}&-{3}\\{32}&{19}&-{47}\end{array}\right|}\)

\(\displaystyle={5}{\left|\begin{array}{cc} {1}&-{3}\\{19}&-{47}\end{array}\right|}-{\left(-{3}\right)}{\left|\begin{array}{cc} {13}&-{3}\\{32}&-{47}\end{array}\right|}+{7}{\left|\begin{array}{cc} {13}&{1}\\{32}&{19}\end{array}\right|}\)

\(\displaystyle={5}{\left[{1}\times{\left(-{47}\right)}-{\left(-{3}\right)}\times{19}\right]}+{3}{\left[{13}\times{\left(-{47}\right)}-{\left(-{3}\right)}\times{32}\right]}+{7}{\left({13}\times{19}-{1}\times{32}\right)}\)

=5(-47+57)+3(-611+96)+7(247-32)

\(\displaystyle={5}\times{10}+{3}\times{\left(-{515}\right)}+{7}\times{215}\)

\(\displaystyle{D}_{{x}}={50}-{1545}+{1505}\)

=10

D=0 and \(\displaystyle{D}_{{x}}\ne{0}\).

Hence, the system of equations is inconsistent.

Hence, the system of equations has no solution.