# 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

Dolly Robinson 2020-12-09 Answered
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
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## Expert Answer

Anonym
Answered 2020-12-10 Author has 108 answers
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:
$\left[\begin{array}{cccc}2& -3& 7& 5\\ 3& 1& -3& 13\\ 2& 19& -47& 32\end{array}\right]$
Step 2
$D=|\begin{array}{ccc}2& -3& 7\\ 3& 1& -3\\ 2& 19& -47\end{array}|$
$=2|\begin{array}{cc}1& -3\\ 19& -47\end{array}|-\left(-3\right)|\begin{array}{cc}3& -3\\ 2& -47\end{array}|+7|\begin{array}{cc}3& 1\\ 2& 19\end{array}|$
$=2\left[1×\left(-47\right)-\left(-3\right)×19\right]+3\left[3×\left(-47\right)-\left(-3\right)×2\right]+7\left(3×19-1×2\right)$
=2(-47+57)+3(-141+6)+7(57-2)
$=2×10+3×\left(-135\right)+7×55$
D=20 - 405 + 385
=0
Step 3
${D}_{x}=|\begin{array}{ccc}5& -3& 7\\ 13& 1& -3\\ 32& 19& -47\end{array}|$
$=5|\begin{array}{cc}1& -3\\ 19& -47\end{array}|-\left(-3\right)|\begin{array}{cc}13& -3\\ 32& -47\end{array}|+7|\begin{array}{cc}13& 1\\ 32& 19\end{array}|$
$=5\left[1×\left(-47\right)-\left(-3\right)×19\right]+3\left[13×\left(-47\right)-\left(-3\right)×32\right]+7\left(13×19-1×32\right)$
=5(-47+57)+3(-611+96)+7(247-32)
$=5×10+3×\left(-515\right)+7×215$
${D}_{x}=50-1545+1505$
=10
D=0 and ${D}_{x}\ne 0$.
Hence, the system of equations is inconsistent.
Hence, the system of equations has no solution.
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Jeffrey Jordon
Answered 2021-11-08 Author has 2064 answers

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