Answer
$(1,1,-2)$
Work Step by Step
Start with the augmented matrix, row reduce to
reduced row echelon form (Gauss-Jordan.)
$\left[\begin{array}{llll}
10 & 10 & -20 & 60\\
15 & 20 & 30 & -25\\
-5 & 30 & -10 & 45
\end{array}\right]\rightarrow\left(\begin{array}{l}
\div 10\\
-1.5R_{1}\\
+0.5R_{1}
\end{array}\right)$
$\rightarrow\left[\begin{array}{llll}
1 & 1 & -2 & 6\\
0 & 5 & 60 & -115\\
0 & 35 & -20 & 75
\end{array}\right]\rightarrow\left(\begin{array}{l}
.\\
\div 5\\
-7R_{2}
\end{array}\right)$
$\rightarrow\left[\begin{array}{llll}
1 & 1 & -2 & 6\\
0 & 1 & 12 & -23\\
0 & 0 & -440 & 880
\end{array}\right]\rightarrow\left(\begin{array}{l}
-R_{2}.\\
.\\
\div(-440)
\end{array}\right)$
$\rightarrow\left[\begin{array}{llll}
1 & 0 & -14 & 29\\
0 & 1 & 12 & -23\\
0 & 0 & 1 & -2
\end{array}\right]\rightarrow\left(\begin{array}{l}
+14R_{3}.\\
-12R_{3}.\\
.
\end{array}\right)$
$\rightarrow\left[\begin{array}{llll}
1 & 0 & 0 & 1\\
0 & 1 & 0 & 1\\
0 & 0 & 1 & -2
\end{array}\right]$
The solution is $(1,1,-2)$
