Questions: Participation Activity #7 This is similar to Try It #1 in the OpenStax text. Solve the system of equations in three variables. 2x + y - 2z = -2 3x - 3y - z = 10 x - 2y + 3z = 12 Enter the exact answer as an ordered triple, (x, y, z). Include a multiplication sign between symbols. For example, a * x.

Solution Steps

To solve the system of equations with three variables, we can use a method such as substitution, elimination, or matrix operations. Here, we'll use the matrix method by representing the system as a matrix equation \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the column matrix of variables, and \(B\) is the constant matrix. We can then solve for \(X\) by finding the inverse of \(A\) and multiplying it by \(B\).

We are given the following system of equations: \[ \begin{align*}

1. & \quad 2x + y - 2z = -2 \\
2. & \quad 3x - 3y - z = 10 \\
3. & \quad x - 2y + 3z = 12 \end{align*} \]

We can represent the system as a matrix equation \(AX = B\), where: \[ A = \begin{bmatrix} 2 & 1 & -2 \\ 3 & -3 & -1 \\ 1 & -2 & 3 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} -2 \\ 10 \\ 12 \end{bmatrix} \]

By solving the matrix equation, we find: \[ X = \begin{bmatrix} 2 \\ -2 \\ 2 \end{bmatrix} \] This gives us the values: \[ x = 2, \quad y = -2, \quad z = 2 \]

Final Answer