Let's proceed **step by step** as requested: --- ## **Step 1: Write the i-system** The given system (i-system) is: \[ \begin{align*} x_1 + 3x_2 &\leq 18 \\ 5x_1 + 4x_2 &\leq 35 \\ x_1, x_2 &\geq 0 \end{align*} \] --- ## **Step 2: Convert to e-system (standard form) using slack variables** We introduce **slack variables** \( s_1 \) and \( s_2 \) to convert the inequalities into equalities: \[ \begin{align*} x_1 + 3x_2 + s_1 &= 18 \qquad (s_1 \geq 0) \\ 5x_1 + 4x_2 + s_2 &= 35 \qquad (s_2 \geq 0) \\ x_1, x_2, s_1, s_2 &\geq 0 \end{align*} \] --- ## **Step 3: List all basic solutions** For a system of \( m \) equations with \( n \) variables, a **basic solution** is found by setting \( n-m \) variables to zero and solving for the rest. - Here, \( m = 2 \) equations, \( n = 4 \) variables (\( x_1, x_2, s_1, s_2 \)) - So, in each basic solution, set \( 4-2 = 2 \) variables to zero, solve for the remaining 2. ### **Possible combinations:** Let’s enumerate all possible pairs of variables to be considered as **basic variables** (solved for), while the other two are set to zero. The variables: \( x_1, x_2, s_1, s_2 \) The possible basic variable pairs: 1. \( x_1, x_2 \) 2. \( x_1, s_1 \) 3. \( x_1, s_2 \) 4. \( x_2, s_1 \) 5. \( x_2, s_2 \) 6. \( s_1, s_2 \) Let’s find the solution for each. --- ## **Step 4: Compute all basic solutions** We will set the non-basic variables to zero and solve for the basic variables in each case. ### **Case 1: Basic variables \( x_1, x_2 \), set \( s_1 = s_2 = 0 \)** \[ \begin{align*} x_1 + 3x_2 = 18 \\ 5x_1 + 4x_2 = 35 \end{align*} \] Solve: Multiply first equation by 5: \[ 5x_1 + 15x_2 = 90 \] Subtract the second equation: \[ (5x_1 + 15x_2) - (5x_1 + 4x_2) = 90 - 35 \implies 11x_2 = 55 \implies x_2 = 5 \] Substitute into first equation: \[ x_1 + 3(5) = 18 \implies x_1 = 18 - 15 = 3 \] So: - \( x_1 = 3 \) - \( x_2 = 5 \) - \( s_1 = 0 \) - \( s_2 = 0 \) Check non-negativity: all variables \(\geq 0\), so **feasible**. --- ### **Case 2: Basic variables \( x_1, s_1 \), set \( x_2 = s_2 = 0 \)** \[ \begin{align*} x_1 + 3(0) + s_1 = 18 \implies x_1 + s_1 = 18 \\ 5x_1 + 4(0) + 0 = 35 \implies 5x_1 = 35 \implies x_1 = 7 \end{align*} \] Plug back: \[ 7 + s_1 = 18 \implies s_1 = 11 \] So: - \( x_1 = 7 \) - \( x_2 = 0 \) - \( s_1 = 11 \) - \( s_2 = 0 \) Check non-negativity: all variables \(\geq 0\), so **feasible**. --- ### **Case 3: Basic variables \( x_1, s_2 \), set \( x_2 = s_1 = 0 \)** \[ \begin{align*} x_1 + 3(0) + 0 = 18 \implies x_1 = 18 \\ 5x_1 + 4(0) + s_2 = 35 \implies 5(18) + s_2 = 35 \implies 90 + s_2 = 35 \implies s_2 = 35 - 90 = -55 \end{align*} \] So: - \( x_1 = 18 \) - \( x_2 = 0 \) - \( s_1 = 0 \) - \( s_2 = -55 \) Check non-negativity: \( s_2 < 0 \), so **not feasible**. --- ### **Case 4: Basic variables \( x_2, s_1 \), set \( x_1 = s_2 = 0 \)** \[ \begin{align*} 0 + 3x_2 + s_1 = 18 \implies 3x_2 + s_1 = 18 \\ 5(0) + 4x_2 + 0 = 35 \implies 4x_2 = 35 \implies x_2 = 8.75 \end{align*} \] Plug back: \[ 3(8.75) + s_1 = 18 \implies 26.25 + s_1 = 18 \implies s_1 = 18 - 26.25 = -8.25 \] So: - \( x_1 = 0 \) - \( x_2 = 8.75 \) - \( s_1 = -8.25 \) - \( s_2 = 0 \) Check non-negativity: \( s_1 < 0 \), so **not feasible**. --- ### **Case 5: Basic variables \( x_2, s_2 \), set \( x_1 = s_1 = 0 \)** \[ \begin{align*} 0 + 3x_2 + 0 = 18 \implies 3x_2 = 18 \implies x_2 = 6 \\ 5(0) + 4x_2 + s_2 = 35 \implies 4(6) + s_2 = 35 \implies 24 + s_2 = 35 \implies s_2 = 11 \end{align*} \] So: - \( x_1 = 0 \) - \( x_2 = 6 \) - \( s_1 = 0 \) - \( s_2 = 11 \) Check non-negativity: all variables \(\geq 0\), so **feasible**. --- ### **Case 6: Basic variables \( s_1, s_2 \), set \( x_1 = x_2 = 0 \)** \[ \begin{align*} 0 + 0 + s_1 = 18 \implies s_1 = 18 \\ 5(0) + 4(0) + s_2 = 35 \implies s_2 = 35 \end{align*} \] So: - \( x_1 = 0 \) - \( x_2 = 0 \) - \( s_1 = 18 \) - \( s_2 = 35 \) Check non-negativity: all variables \(\geq 0\), so **feasible**. --- ## **Step 5: Summary Table of All Basic Solutions** | Case | Basic Vars | \( x_1 \) | \( x_2 \) | \( s_1 \) | \( s_2 \) | Feasible? | |------|--------------|-----------|-----------|-----------|-----------|-----------| | 1 | \( x_1, x_2 \) | 3 | 5 | 0 | 0 | Yes | | 2 | \( x_1, s_1 \) | 7 | 0 | 11 | 0 | Yes | | 3 | \( x_1, s_2 \) | 18 | 0 | 0 | -55 | No | | 4 | \( x_2, s_