# Investigate for consistency of the following equations and if possible then find the solutions: 2x-3y+7z=5 3x+y-3z=13 2x+19y-47z=32

Question
Equations
Investigate for consistency of the following equations and if possible then find the solutions:
2x-3y+7z=5
3x+y-3z=13
2x+19y-47z=32

2020-12-10
Step 1
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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.

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