Given the system of equations: \n 1. \(2x - y + 3z = 7\)\n 2. \(x + 2y - z = 6\)\n 3. \(3x - y + 2z = 10\)\n Rewrite the system as a vector equality.
Solution
Step 1 ：First, we can express each equation in terms of vectors. \n For equation 1, the vector is \(\begin{bmatrix} 2 \ -1 \ 3 \end{bmatrix}^T\). \n For equation 2, the vector is \(\begin{bmatrix} 1 \ 2 \ -1 \end{bmatrix}^T\). \n For equation 3, the vector is \(\begin{bmatrix} 3 \ -1 \ 2 \end{bmatrix}^T\).
Step 2 ：Next, we can express \(x\), \(y\), and \(z\) as a vector, which is \(\begin{bmatrix} x \ y \ z \end{bmatrix}^T\).
Step 3 ：Then, we can express the constants on the right side of each equation as a vector, which is \(\begin{bmatrix} 7 \ 6 \ 10 \end{bmatrix}^T\).
Step 4 ：Finally, we can rewrite the system of equations as a vector equation. This equation is \(\begin{bmatrix} 2 & -1 & 3 \ 1 & 2 & -1 \ 3 & -1 & 2 \end{bmatrix}\begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 7 \ 6 \ 10 \end{bmatrix}\)
