To solve the given system of equations:
[tex]\[ \left\{\begin{array}{l}
-3x + 4y + 5z = 13 \\
-6x - 4y - 2z = 14 \\
12x - 4y + 2z = -10
\end{array}\right. \][/tex]
we will employ the method of row reduction to bring the augmented matrix to its reduced row echelon form (RREF).
### Step 1: Formulate the augmented matrix
We begin by writing the augmented matrix for the system of equations:
[tex]\[
\begin{pmatrix}
-3 & 4 & 5 & | & 13 \\
-6 & -4 & -2 & | & 14 \\
12 & -4 & 2 & | & -10
\end{pmatrix}
\][/tex]
### Step 2: Perform row operations to obtain RREF
After performing several row operations to bring it to RREF, we end up with:
[tex]\[
\begin{pmatrix}
1 & 0 & 0 & | & -2 \\
0 & 1 & 0 & | & -2 \\
0 & 0 & 1 & | & 3
\end{pmatrix}
\][/tex]
### Step 3: Interpret the RREF matrix
Each row in the RREF corresponds to the following equations:
1. [tex]\(1x + 0y + 0z = -2\)[/tex]
2. [tex]\(0x + 1y + 0z = -2\)[/tex]
3. [tex]\(0x + 0y + 1z = 3\)[/tex]
From these equations, we directly read off the values for [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex]:
[tex]\[
x = -2
\][/tex]
[tex]\[
y = -2
\][/tex]
[tex]\[
z = 3
\][/tex]
### Conclusion
The solution to the system of equations is the unique ordered triple [tex]\( (x, y, z) = (-2, -2, 3) \)[/tex].
