Let's break down the problem step by step: ### **Step 1: Define Variables** Let - \( x \) = number of chairs produced per day - \( y \) = number of sofas produced per day - \( z \) = number of tables produced per day ### **Step 2: Write the Objective Function** Profit per chair = $70 Profit per sofa = $50 Profit per table = $100 **Objective:** Maximize profit \[ P = 70x + 50y + 100z \] ### **Step 3: Write the Constraints** From the table: - **Carpentry:** \( 6x + 3y + 8z \leq 720 \) - **Finishing:** \( 1x + 1y + 2z \leq 148 \) - **Upholstery:** \( 4x + 6y + 0z \leq 288 \) - **Non-negativity:** \( x \geq 0, \ y \geq 0, \ z \geq 0 \) --- ### **Step 4: Setup the System for Guess-Jordan Method** The **Gauss-Jordan method** is primarily used for solving systems of linear equations, not inequalities. However, you can use it to find the intersection points (corner points) of the constraints (i.e., treat constraints as equations), and then check which are feasible. #### Let's convert constraints to equalities to find corner points: \[ \begin{align*} 6x + 3y + 8z &= 720 \quad \text{(1)} \\ x + y + 2z &= 148 \quad \text{(2)} \\ 4x + 6y &= 288 \quad \text{(3)} \\ \end{align*} \] --- ### **Step 5: Solve the System (Gauss-Jordan Method)** Let’s solve for intersections: #### **A. Use Equation (3) to Solve for x in terms of y:** \[ 4x + 6y = 288 \implies x = \frac{288 - 6y}{4} = 72 - 1.5y \] #### **B. Substitute into Equation (2):** \[ x + y + 2z = 148 \\ (72 - 1.5y) + y + 2z = 148 \\ 72 - 0.5y + 2z = 148 \\ -0.5y + 2z = 76 \\ 0.5y = 2z - 76 \\ y = 4z - 152 \] #### **C. Substitute x and y into Equation (1):** Recall: - \( x = 72 - 1.5y \) - \( y = 4z - 152 \) \[ 6x + 3y + 8z = 720 \\ 6(72 - 1.5y) + 3y + 8z = 720 \\ 432 - 9y + 3y + 8z = 720 \\ 432 - 6y + 8z = 720 \\ -6y + 8z = 288 \\ -6y = 288 - 8z \\ y = \frac{8z - 288}{6} = \frac{4z - 144}{3} \] But we also have \( y = 4z - 152 \). Set equal: \[ 4z - 152 = \frac{4z - 144}{3} \\ 3(4z - 152) = 4z - 144 \\ 12z - 456 = 4z - 144 \\ 12z - 4z = -144 + 456 \\ 8z = 312 \\ z = 39 \] Now, \[ y = 4z - 152 = 4(39) - 152 = 156 - 152 = 4 \] \[ x = 72 - 1.5y = 72 - 1.5 \times 4 = 72 - 6 = 66 \] --- ### **Step 6: Check Feasibility** Plug \( x = 66, y = 4, z = 39 \) into the original inequalities: 1. **Carpentry:** \( 6(66) + 3(4) + 8(39) = 396 + 12 + 312 = 720 \leq 720 \) ✔️ 2. **Finishing:** \( 66 + 4 + 2(39) = 66 + 4 + 78 = 148 \leq 148 \) ✔️ 3. **Upholstery:** \( 4(66) + 6(4) + 0 = 264 + 24 = 288 \leq 288 \) ✔️ All are satisfied. --- ### **Step 7: Calculate Maximum Profit** \[ P = 70x + 50y + 100z = 70(66) + 50(4) + 100(39) = 4620 + 200 + 3900 = \boxed{8720} \] --- ## **Final Answer** **To maximize profit, the manufacturer should produce:** - **66 chairs** - **4 sofas** - **39 tables** **Maximum profit:** \(\boxed{\$8720}\) --- ### **Note:** The Gauss-Jordan method solves systems of linear equations (equalities), which helps find the corner points (vertices) of the feasible region in linear programming. The maximum profit in such problems will always occur at one of these corner points. If you want to check other combinations (like where \( x \) or \( y \) or \( z \) is zero), repeat similar steps for those cases. But in this setup, the above point is the optimal one.