Answer
The system of equations has no solution
Work Step by Step
The provided system of equations is:
$\begin{align}
5x+12y+z=10 & \\
2x+5y+2z=-1 & \\
x+2y-3z=5 & \\
\end{align}$
The given system of equations can be written in matrix form as below: $\left[ \left. \begin{matrix}
5 & 12 & 1 \\
2 & 5 & 2 \\
1 & 2 & -3 \\
\end{matrix} \right|\begin{matrix}
10 \\
-1 \\
5 \\
\end{matrix} \right]$
Now solve this matrix.
$\left[ \left. \begin{matrix}
5 & 12 & 1 \\
0 & 1 & 8 \\
1 & 2 & -3 \\
\end{matrix} \right|\begin{matrix}
10 \\
-11 \\
5 \\
\end{matrix} \right]$ $ By,\ {{R}_{2}}-2{{R}_{3}}\to {{R}_{2}}$
$\left[ \left. \begin{matrix}
5 & 12 & 1 \\
0 & 1 & 8 \\
0 & 2 & 16 \\
\end{matrix} \right|\begin{matrix}
10 \\
-11 \\
-15 \\
\end{matrix} \right]$ $ By,\ {{R}_{1}}-5{{R}_{3}}\to {{R}_{3}}$
$\left[ \left. \begin{matrix}
5 & 12 & 1 \\
0 & 1 & 8 \\
0 & 0 & 0 \\
\end{matrix} \right|\begin{matrix}
10 \\
-11 \\
-7 \\
\end{matrix} \right]$ $ By,\ 2{{R}_{2}}-{{R}_{3}}\to {{R}_{3}}$
Convert the last row in equation form:
$\begin{align}
0x+0y+0z=-7 & \\
0=-7 & \\
\end{align}$
Which is not possible.
