Determine whether \( (0,2,1,-2) \) satisfies each equation in the following system of equations. Equation Satisfies? \[ \begin{array}{ll}5 x_{1}+2 x_{2}+2 x_{3}+x_{4}=5 & (1) \\ 6 x_{1}+3 x_{2}+2 x_{3}+x_{4}=5 & (2) \\ 6 x_{1}+3 x_{2}+3 x_{3}+x_{4}=6 & (3) \\ 4 x_{1}+2 x_{2}+2 x_{3}+x_{4}=5 & (4) \\ \text { Is }(0,2,1,-2) \text { a solution of the system of equations? } \\ \text { V Select } \\ \text { No } \\ \text { Yes }\end{array} \]

To determine whether the point \( (0, 2, 1, -2) \) satisfies each equation in the given system, we will substitute \( x_1 = 0 \), \( x_2 = 2 \), \( x_3 = 1 \), and \( x_4 = -2 \) into each equation and check if the left-hand side equals the right-hand side. Let's evaluate each equation step by step: 1. **Equation (1):** \[ 5 x_{1} + 2 x_{2} + 2 x_{3} + x_{4} = 5 \] Substituting the values: \[ 5(0) + 2(2) + 2(1) + (-2) = 0 + 4 + 2 - 2 = 4 \neq 5 \] **Does not satisfy.** 2. **Equation (2):** \[ 6 x_{1} + 3 x_{2} + 2 x_{3} + x_{4} = 5 \] Substituting the values: \[ 6(0) + 3(2) + 2(1) + (-2) = 0 + 6 + 2 - 2 = 6 \neq 5 \] **Does not satisfy.** 3. **Equation (3):** \[ 6 x_{1} + 3 x_{2} + 3 x_{3} + x_{4} = 6 \] Substituting the values: \[ 6(0) + 3(2) + 3(1) + (-2) = 0 + 6 + 3 - 2 = 7 \neq 6 \] **Does not satisfy.** 4. **Equation (4):** \[ 4 x_{1} + 2 x_{2} + 2 x_{3} + x_{4} = 5 \] Substituting the values: \[ 4(0) + 2(2) + 2(1) + (-2) = 0 + 4 + 2 - 2 = 4 \neq 5 \] **Does not satisfy.** Now, let's summarize the results: - Equation (1): Does not satisfy - Equation (2): Does not satisfy - Equation (3): Does not satisfy - Equation (4): Does not satisfy Since \( (0, 2, 1, -2) \) does not satisfy any of the equations in the system, the answer is: **No, \( (0, 2, 1, -2) \) is not a solution of the system of equations.**