Determine which of the following is the solution set of the linear equations below. II \( 3 x-y+z=2 \) \( 2 x-z=2 \) \( \{(x, y, z): x=t, y=5 t+4, z=2 t-2 \) with \( t \in \mathbb{R}\} \) \( \left\{(x, y, z): x=\frac{1}{3}(t+s+2), y=t, z=s\right. \) with \( \left.s, t \in \mathbb{R}\right\} \) \( \left\{(x, y, z): x=-\frac{1}{3}(2+t-s), y=t, z=s\right. \) with \( \left.s, t \in \mathbb{R}\right\} \) \( \left\{(x, y, z): x=\frac{1}{3}(t-s-2), y=t, z=s\right. \) with \( \left.s, t \in \mathbb{R}\right\} \) None of the given option (s) \( \left\{(x, y, z): x=\frac{1}{3}(t-s+2), y=t, z=s\right. \) with \( \left.s, t \in \mathbb{R}\right\} \)

To determine which of the given options is the solution set of the linear equations: 1. \( 3x - y + z = 2 \) 2. \( 2x - z = 2 \) we can start by solving the system of equations. From the second equation, we can express \( z \) in terms of \( x \): \[ z = 2x - 2 \] Now, we can substitute this expression for \( z \) into the first equation: \[ 3x - y + (2x - 2) = 2 \] Simplifying this gives: \[ 3x - y + 2x - 2 = 2 \] \[ 5x - y - 2 = 2 \] \[ 5x - y = 4 \] Now we can express \( y \) in terms of \( x \): \[ y = 5x - 4 \] Now we have \( y \) and \( z \) in terms of \( x \): \[ y = 5x - 4 \] \[ z = 2x - 2 \] We can express the solution set in parametric form by letting \( x = t \): \[ x = t, \quad y = 5t - 4, \quad z = 2t - 2 \] Thus, the solution set can be written as: \[ \{(x, y, z): x = t, y = 5t - 4, z = 2t - 2 \text{ with } t \in \mathbb{R}\} \] Now, we can compare this with the given options: 1. \( \{(x, y, z): x=t, y=5t+4, z=2t-2 \} \) - Incorrect (the \( y \) term is wrong) 2. \( \left\{(x, y, z): x=\frac{1}{3}(t+s+2), y=t, z=s\right. \) - Incorrect 3. \( \left\{(x, y, z): x=-\frac{1}{3}(2+t-s), y=t, z=s\right. \) - Incorrect 4. \( \left\{(x, y, z): x=\frac{1}{3}(t-s-2), y=t, z=s\right. \) - Incorrect 5. None of the given option (s) - This could be true 6. \( \left\{(x, y, z): x=\frac{1}{3}(t-s+2), y=t, z=s\right. \) - Incorrect Since none of the options match the derived solution set \( \{(x, y, z): x = t, y = 5t - 4, z = 2t - 2 \text{ with } t \in \mathbb{R}\} \), the correct answer is: **None of the given option (s)**.