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

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:
${\displaystyle \left[\begin{array}{cccc}2& -3& 7& 5\\ 3& 1& -3& 13\\ 2& 19& -47& 32\end{array}\right]}$
Step 2
${\displaystyle D=\left|\begin{array}{ccc}2& -3& 7\\ 3& 1& -3\\ 2& 19& -47\end{array}\right|}$
${\displaystyle =2\left|\begin{array}{cc}1& -3\\ 19& -47\end{array}\right|-(-3)\left|\begin{array}{cc}3& -3\\ 2& -47\end{array}\right|+7\left|\begin{array}{cc}3& 1\\ 2& 19\end{array}\right|}$
${\displaystyle =2[1\times (-47)-(-3)\times 19]+3[3\times (-47)-(-3)\times 2]+7(3\times 19-1\times 2)}$
=2(-47+57)+3(-141+6)+7(57-2)
${\displaystyle =2\times 10+3\times (-135)+7\times 55}$
D=20 - 405 + 385
=0
Step 3
${\displaystyle D_x=\left|\begin{array}{ccc}5& -3& 7\\ 13& 1& -3\\ 32& 19& -47\end{array}\right|}$
${\displaystyle =5\left|\begin{array}{cc}1& -3\\ 19& -47\end{array}\right|-(-3)\left|\begin{array}{cc}13& -3\\ 32& -47\end{array}\right|+7\left|\begin{array}{cc}13& 1\\ 32& 19\end{array}\right|}$
${\displaystyle =5[1\times (-47)-(-3)\times 19]+3[13\times (-47)-(-3)\times 32]+7(13\times 19-1\times 32)}$
=5(-47+57)+3(-611+96)+7(247-32)
${\displaystyle =5\times 10+3\times (-515)+7\times 215}$
${\displaystyle D_x=50-1545+1505}$
=10
D=0 and ${\displaystyle D_x\ne 0}$.
Hence, the system of equations is inconsistent.
Hence, the system of equations has no solution.